Physics - Oak Park Independent



Physics 5: Circular and Rotational Motion Name __________________________

A. Circular Motion (5-1 to 5-3, 5-6 to 5-8)

1. constant perimeter (tangential) speed: vt = 2πr/T

a. distance = circumference of the circle: 2πr

b. time = time for one revolution: T (period)

2. constant inward (centripetal) acceleration: ac = v2/r

3. centripetal force, Fc = mac = mv2/r

a. turning on a road problems

| v = 2πr/T |

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|ac |

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|when the road is horizontal: Fc = Ff = μsmg |

|roads are banked in order to reduce the amount of friction (component of|

|the Fg is || to Fc) |

b. horizontal loop problem (mass on a string)

| Ft-x = Fc = mv2/r |

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|θ |

|Ft Ft-y = Fg = mg |

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|v = 2πr/T |

|Ft = (Fc2 + Fg2)½ |

|tanθ = Fg/Fc (θ is measured from horizontal) |

c. vertical loop problem (mass on a string)

|top: Fnet = Fc = Ft + Fg ∴ Ft = Fc – Fg |

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|Fg Ft |

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|Fg Ft |

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|bottom: Fnet = Fc = Ft – Fg ∴ Ft = Fc + Fg |

|if on a roller coaster: Fn = Ft |

4. Newton's law of universal gravity, Fg = GMm/r2

a. G = 6.67 x 10-11 N•m2/kg2

b. M = mplanet and m = msatellite

c. r is the distance, measured from center to center

d. g = GM/r2

e. Fg = Fc: GMm/r2 = mv2/r ∴ v = (GM/r)½

v = 2πr/T

m Fg = Fg M

r

| |Mass (kg) |Radius (m) |r from Earth (m) |

|Earth |5.98 x 1024 |6.38 x 106 | |

|Moon |7.35 x 1022 |1.74 x 106 |3.84 x 108 |

|Sun |1.99 x 1030 |6.96 x 108 |1.50 x 1011 |

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B. Newton's Laws—Rotation (8-3 to 8-6)

1. torque, τ = r⊥Fr (tau—Greek letter for t)

rotation

r 90o Fr

a. r = perpendicular distance from axis of rotation to rotating force Fr

b. when r is not perpendicular to Fr, then τ = rFrsinθ

c. torque units are m•N (not N•m—work)

2. First Law: Object remains at rest or uniform rotation as long as no net torque (τnet) acts on it

a. measured as the moment of inertia, I = βmr2

b. β corrects for mass distribution (β = 1 for a hoop)

c. equilibrium (τnet = 0)

1. center of mass for a complex system—balance point where an upward torque generated by a force equal to the total weight counters the sum of all the downward weight torques

|rcm Fcm = (m1 + m2 + m3)g |

|m1 m2 |

|m3 |

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|r1 F1 |

|r2 F2 |

|r3 |

|F3 |

|not rotating ∴ τCM = τ1 + τ2 + τ3 |

|rcm(m1 + m2 + m3)g = r1m1g + r2m2g + r3m3g |

|rcm = (r1m1 + r2m2 + r3m3)/(m1 + m2 + m3) |

|rcm = Σ(rimi)/Σmi |

2. cantilever problems—how far from the edge can m2 be placed without tipping?

| mass of plank at its center |

|m1 m2 |

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|r1 r2 |

|Fg1 Fg2 |

|not rotating ∴ τ1 = τ2 |

|r1m1g = r2m2g |

|r1m1 = r2m2 |

3. two supports problems—what are the tensions?

| FL mass of plank at its center FR |

|rR |

|m1 m2 |

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|r1 |

|r2 |

|assume left side is point of rotation |

|not rotating ∴ τR = τ1 + τ2 |

|rRFR = r1m1g + r2m2g |

|FR = (r1m1g + r2m2g)/rR |

|solve for FL: FL + FR = m1g + m2g |

3. Second Law: τnet = Iα (τ = rFr, Ι = βmr2, α = a/r)

a. rFr = (βmr2)(a/r) ∴ Fr = βma (acceleration at rim)

b. pulley problems—what is the acceleration

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|m1 |

|rough surface (μ) m2 |

|Fp – Ff = m1a + βm2a β |

|Fp – μm1g = (m1 + βm2)a |

|solve for d, v or t using kinematics Fp|

c. rolling problems—what is the acceleration

| β Frolling = ma + βma |

|m Frolling = (1 + β)ma |

|Fgsinθ = (1 + β)ma |

|solve for d, v or t using kinematics |

|Fgsinθ |

|θ |

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C. Conservation Laws—Rotation (8-7 to 8-8)

1. rotational momentum, L = rp = rβmv (kg•m2/s)

a. when τnet = 0, then ΔL = 0

b. change r and/or β will change v

orbiting planet spinning diver

Kepler's Law

(A1 → 2 = A3 → 4)

r1v1 = r2v2 r1β1v1 = r2β2v2

2. rotational kinetic energy, Kr = ½βmv2 (J)

a. v is the velocity at the rim

b. rolling kinetic energy: Krolling = ½(1 + β)mv2

3. mixed linear and rotation motion problems

a. Summary of translational and rotational formulas

|Variable |Translational |Rotational |Rolling |

|force |F = ma |Fr = βma |F = (1 + β)ma |

|momentum |p = mv |L = rβmv |p + L = (1 + rβ)mv |

|kinetic energy |K = ½mv2 |Kr = ½βmv2 |K = ½(1 + β)mv2 |

b. collision problems (conservation of momentum)—what is the velocity after the boy jumps on the merry-go-round?

| m v |

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|convert linear into rotational (r = rM, β = 1) |

|Lm + LM = L' M, rM, βM |

|rMβmmvm + rMβMMvM = (rMβmm + rMβMM)v' |

|mvm = (mm + βMM)v' |

|v' = mvm/(mm + βMM) |

c. pulley problems (conservation of energy)—what is the velocity after descending h meters?

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|m1 |

|rough surface (μ) m3 |

|Ug3 – Wf1 = K1 + K2 + K3 β |

|m3gx – μkm1gx = ½(m1 + m2 + βm3)v'2 |

|v' = [2(m3 – μkm1)gx/(m1 + m2 + βm3)]½ |

|m2 |

d. pulley problems (conservation of energy)—what is the velocity after descending h meters?

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|Ug1 + Ug2 = Ug1' + Ug2' + K1 + K2 + K3 m3 |

|0 + m2gh = m1gh + 0 + ½(m1 + m2 + βm3)v'2 β |

|(m2 – m1)gh = ½(m1 + m2 + βm3)v'2 |

|v = [2(m2 – m1)gh/(m1 + m2 + βm3)]½ |

|m2 |

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|m1 h |

e. rolling problems (conservation of energy)—what is the velocity after descending h meters?

| m Ug = Krolling |

|β mgh = ½(1 + β)mv2 |

|h gh = ½(1 + β)v2 |

|v = [2gh/(1 + β)]½ |

f. cart problems (conservation of energy)—what is the velocity after descending h meters?

| m1 Ug = K + Krolling |

|? ? β, m2 (m1 + 4m2)gh = ½m1v2 + ½(1 + β)4m2v2 |

|(m1 + 4m2)gh = ½(m1 + (1 + β)4m2)v2 |

|h |

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D. Simple Harmonic Motion (SHM) (11-1 to 11-6)

1. oscillating mass on a spring

a. acceleration is NOT constant ∴ kinematic formulas are invalid

b. displacement, velocity and acceleration oscillate between +A and –A, where A = amplitude

1. x = +A, when t = 0 (pictured above)

|time |t = 0 |t = ¼T |t = ½T |t = ¾T |t = T |

|displacement |+A |0 |-A |0 |+A |

|velocity |0 |-vmax |0 |+vmax |0 |

|acceleration |-amax |0 |+amax |0 |-amax |

2. x = 0, when t = 0 (heading downward)

|time |t = 0 |t = ¼T |t = ½T |t = ¾T |t = T |

|displacement |0 |-A |0 |+A |0 |

|velocity |-vmax |0 |+vmax |0 |-vmax |

|acceleration |0 |+amax |0 |-amax |0 |

c. maximum acceleration, amax = A(k/m) = vmax2/A

|Steps |Algebra |

|start with |Fs = ma |

|substitute kA for Fs |kA = ma |

|solve for a |amax = A(k/m) |

d. maximum velocity, vmax = A(k/m)½ = 2πA/T

|Steps |Algebra |

|start with |Us = K |

|substitute ½kA2 for Us and ½mv2 for K |½kA2 = ½mv2 |

|solve for v |vmax = A(k/m)½ |

e. time for one cycle, period, T = 2π(m/k)½

|Steps |Algebra |

|start with |vmax = A(k/m)½ |

|substitute 2πA/T for vmax |2πA/T = A(k/m)½ |

|simplify |2π/T = (k/m)½ |

|solve for T |T = 2π/(k/m)½ |

|substitute (m/k)½ for 1/(k/m)½ |T = 2π(m/k)½ |

f. formulas at midpoint, 0, and extremes, A

| |midpoint |extreme |

|x |0 |xmax = A |

|v |vmax = A(k/m)½ = 2πA/T |0 |

|a |0 |amax = -A(k/m) = vmax2/A |

|F |0 |F = -ma = -kA |

|U |0 |Umax = ½kA2 |

|K |Kmax = ½mv2 |0 |

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2. pendulum

[pic]

a. period of the simple pendulum, T = 2π(L/g)½

|Steps |Algebra |

|start with |F = kx |

|substitute mgsinθrad for F |mgsinθrad = kx |

|substitute Lθrad for x |mgsinθrad = kLθrad |

|for small angles sinθrad = θrad |mg = kL |

|solve for k |k = mg/L |

|start with |T = 2π(m/k)½ |

|substitute mg/L for k |T = 2π(m/mg/L)½ |

|simplify |T = 2π(L/g)½ |

b. notice that m cancels out of the equation, so the period only depends on the L and g

3. damped harmonic motion

[pic]

a. amplitude of oscillating spring or swinging pendulum will decrease until it stops—damping

b. damping is due to friction and air resistance

1. forces always oppose direction of velocity

2. damping is enhanced if oscillator is placed in viscous fluid (car shock absorbers)

c. forced damping is accomplished with motors that are programmed to oppose velocity (earthquake protected buildings)

4. resonance

a. object can be set to oscillate by an external force—forced vibration

b. when forced vibration matches natural vibration, then amplitude builds with each vibration—resonance

c. examples

1. child swinging

2. building during an earthquake

3. air inside a musical instrument

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Experiments

1. Centripetal Force Lab

a. Measure the time it takes the stopper to make 10 revolutions with the given lengths and hanging masses.

|Control |

|string length, L |0.5 m |

|hanging weight, m1 |200 g |

|stopper mass, m2 | |

|time (10 orbits), t | |

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|Double L |Half m1 |

|m1 |200 g |m1 |100 g |

|string length, L |1.0 m |string length, L |0.5 m |

|time (10 orbits), t | |time (10 orbits), t | |

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| |m1g |m2g |Ftx |Fty |

|What is Fty equal to? | | | | |

|Which force equals Fc? | | | | |

|What is Ft equal to? | | | | |

c. Calculate the following from the data.

| |Control |Double L |Half m1 |

|T1 based on t | | | |

|Ft based on m1 | | | |

|Fty based on m2 | | | |

|θ based on Ft and Fty | | | |

|Fc based on Ft and Fty | | | |

|r based on L and θ | | | |

|v based on Fc | | | |

|T2 based on v | | | |

|%Δ T1 and T2 (use average | | | |

|as standard) | | | |

c. How was the period affected by the following?

|Doubling L | |

|Halving m1 | |

2. Equilibrium Lab

a. Extend from the table edge a ½-m stick with a 50-g mass at 0 cm and measure the balance point (CM).

50 g

½-m stick |← rr → |← r50 →

table | | |

50 cm 25 cm 0 cm

(1) Collect the following data.

|r50 |rr |ruler mass, mr |

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(2) Use the cantilever technique to determine the mass of the ruler.

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(3) Determine the percent difference between the measured mr (true) and the calculated mr from (2).

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b. Extend from the table edge a ½-m stick with 50-g at 40 cm, 10-g at 15 cm and 20-g at 5 cm and measure the balance point (rcm).

50 g 10 g 20 g

mr

½-m stick| | | | |

table 50 40 25 15 5 0 cm

|(1) Record the balance point rcm. | |

(2) Use the center of mass formula to determine rcm.

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(3) Determine the percent difference between the measured rcm (true) and the calculated rcm from (2).

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c. Explore the relationship between center-of-mass and balance by performing the following.

(1) Stand with your heels and back against a wall and try to bend over and touch your toes. Explain

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(2) Stand facing the wall with your toes against the wall and try to stand on your toes. Explain

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(3) Rest a meter stick on two fingers. Slowly bring your fingers together. Explain

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3. 1 + β lab

a. Roll the objects down an incline. Measure the change in height, rolling distance and time it takes the object to roll from the top to the bottom of the incline.

|Ring |Disk |

|height, h | |height, h | |

|distance, d | |distance, d | |

|time, t | |time, t | |

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|height, h | |height, h | |

|distance, d | |distance, d | |

|time, t | |time, t | |

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c. Calculate (1 + β) for using conservation of energy.

|Ring |Disk |Ball |Cart |

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d. Calculate the percent difference with the ideal values.

|Ring |Disk |Ball |Cart |

|2 |1.5 |1.4 |1.0 |

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4. Simple Harmonic Motion Lab

a. Measure the length L and time t for 10 oscillations of a spring with different hanging masses m, determine k using two methods and compare the results.

(1) Collect the following data.

|m (kg) |0 |0.10 |0.20 |0.30 |0.40 |0.50 |

|L (m) | | | | | | |

|t (s) | | | | | |

|Fg |Fg = mg |

(3) Determine the following using time data.

|added mass |0.10 |0.20 |0.30 |0.40 |0.50 |

|T |T = t/10 |

(4) Calculate the percent difference between the two values of k using the average as true.

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b. Measure the pendulum period for different releasing angles θ and see which angle gives the most ideal values for T.

(1) Collect the following data.

|L (m) | |

|θ (o) |5o |10o |15o |20o |25o |

|T1 (s) | | | | | |

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(2) Calculate the following from the data.

| |angle |5o |10o |15o |20o |

(3) Calculate T3 = 2π(L/g)½.

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(4) Which angle produced T closest to T3?

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Practice Problems

A. Circular Motion

1. When a tetherball is whirling around the pole, the net force is directed

(A) toward the top of the pole

(B) toward the ground

(C) horizontally away from the pole

(D) horizontally toward the pole

2. You are standing in a bus that makes a sharp left turn. Which of the following is true?

(A) you lean to the left because of centripetal force

(B) you lean to the right because of inertia

(C) you lean forward because of the net force is forward

(D) you lean to the right because of centrifugal force

3. You drive your car too fast around a curve and the car starts to skid. What is the correct description of this situation?

(A) car's engine is not strong enough to keep the car from being pushed out

(B) friction between the tires and the road is not strong enough to keep the car in a circle

(C) car is too heavy to make the turn

(D) none of the above

4. A steel ball is whirling around in a circle on the end of a string when the string breaks. Which path will it follow?

A B C

5. Two stones A and B have the same mass. They are tied to strings and whirled in horizontal circles. The radius of the circular path for stones A is twice the radius of stone B's path. If the period of motion is the same for both stones, what is the tension in cord A (FA) compared to cord B (FB)?

(A) FA = FB (B) FA = 2FB (C) FA = ½FB

6. You driving along a rural road. Which is true when you are at the lowest point along a dip in the road?

(A) Fn < Fg (B) Fn = Fg (C) Fn > Fg

7. You swing a ball on the end of a string in a vertical circle. Which is true of the centripetal force at the top of the circle?

(A) Fc = Ft + Fg (B) Fc = Ft – Fg (C) Fc = Fg – Ft

8. Which is stronger the Earth's pull on the Moon or the Moon's pull on the Earth?

(A) Earth's pull (B) Moon's pull (C) they are equal

9. Is there a net force acting on an astronaut floating in orbit around the Earth while on a space walk?

(A) yes (B) no

10. A car is traveling clockwise on the north side of a circular track. (r = 50 m) takes 16 s to make one lap. Determine

|v | |

|direction of v | |

|ac | |

|direction of ac | |

11. The earth is 1.5 x 1011 m from the sun and makes one complete circular orbit in 1 year.

a. What is the period of orbit in seconds?

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b. What is the earth’s orbital velocity?

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c. What is the centripetal acceleration of the earth toward the sun?

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12. A driver of a 1000-kg sports car attempts a turn whose radius of curvature is 50 m on a road where μ = 0.8.

a. What is the fastest that the driver can make the turn?

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b. Could the driver make the turn at this speed

(1) with a 2,000-kg SUV? Explain

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(2) when the road is wet? Explain

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13. A 2-kg mass is moving at 5 m/s in a horizontal circle of radius 1 m at the end of a cord.

a. What is the horizontal component of tension?

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b. What is the vertical component of tension?

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c. What is the overall tension in the cord?

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d. What angle does the cord make with the horizontal?

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14. A 2-kg mass is moving at 5 m/s in a vertical circle of radius 1 m at the end of a cord.

a. What is the tension in the cord at the top of the circle?

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b. What is the tension in the cord at the bottom?

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15. A 1-kg pendulum bob swings back and forth from a 2-m string that can support 15 N of tension without breaking.

a. What is the maximum speed that the bob can reach at the bottom of the swing without breaking the string?

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b. What is the maximum height measured from vertical that the bob can reach?

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16. How would the force of gravity be affected if the Earth

a. had the same mass but a smaller radius?

|Between Earth and Moon |On the Earth's surface |

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b. had the same radius but a smaller mass?

|Between Earth and Moon |On the Earth's surface |

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17. Determine the acceleration due to gravity on the planet compared to Earth.

|Mass |Radius (x Earth) |Acceleration (x gEarth) |

|m = mEarth |r = rEarth |g |

|m = mEarth |r = 2rEarth | |

|m = mEarth |r = ½rEarth | |

|m = 2mEarth |r = rEarth | |

|m = ½mEarth |r = rEarth | |

18. What is the acceleration due to gravity (g) on Mars?

( m = 6.4 x 1023 kg, r = 3.4 x 106 m)

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19. When the Apollo Missions went to the moon they passed a point where the gravitational attractions from the moon and the earth are equal. What is the ratio rEarth/rMoon where this happened if mEarth/mMoon = 100?

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20. Consider the following changes to earth.

I Increase earth's mass II Decrease earth's mass

III Increase earth's radius IV Decrease earth's radius

|Which changes would decrease the acceleration due to gravity| |

|on the earth's surface? | |

|Which changes would increase the acceleration due to gravity| |

|on the earth's surface? | |

|Which changes would decrease the acceleration due to gravity| |

|on the moon? | |

|Which changes would increase the acceleration due to gravity| |

|on the moon? | |

21. The Earth's mass is 81 times the mass of the moon and the Earth's radius is 4 times the radius of the moon.

a. What is gMoon in terms of gEarth?

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b. What is the mass of a 50 kg person on the Moon?

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c. What is the weight of a 50 kg person on the Moon?

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B. Newton's Laws—Rotation

22. You are using a wrench to loosen a rusty nut. Which will produce the greatest torque?

A B

C

D

23. 3 identical balls descend 3 identical ramps (except for μs). Ball A slides down ramp A (μs = 0), ball B rolls down ramp B (μs = .3) and ball C rolls down ramp C (μs = .6). Which is true of their velocities when the reach the end of their ramp?

(A) vA > vB = vC (B) vA > vB > vC (C) vA = vB = vC

Questions 24-25 Four objects have the same mass and radius.

F

axis of rotation → ?

(A) hollow cylinder, β = 1 (B) solid cylinder, β = 1/2

(C) hollow ball, β = 2/3 (D) solid ball, β = 2/5

24. Which would have the greatest moment of inertia?

25. Which would have the greatest rotational acceleration?

26. A 1-kg block is hung at the end of a rod 1-m long. The balance point is 0.25 m from the end holding the block, what is the mass of the rod?

|← 0.25 m →|← 0.25 m →|

center of rod

1 kg

(A) 0.25 kg (B) 0.5 kg (C) 1 kg (D) 2 kg

27. What is the total mass of the mobile? (rods are massless)

1 m 2 m A

B 1 m 3 m 1 kg

(A) 5 kg (B) 6 kg (C) 7 kg (D) 8 kg

28. Consider the two configurations of interlocking blocks on the edge of a table. Which of the following is true?

A B

(A) A tips (B) B tips (C) both tip (D) neither tip

29. Consider the door as viewed from above.

Determine

a. The torque when F1 = 45 N and r1 = 1 m.

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b. The force, F2, where r2 = 0.4 m, that will generate the same torque as part a.

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30. A 5-kg disk (β = ½) rolls down a 30o incline. Determine

a. The parallel component of Fg.

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b. The disk's acceleration at the rim.

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31. A 25-kg box rests on the edge of a merry-go-round (r = 2 m).

a. What is the maximum force of friction between the box and merry-go-round (μs = 0.80)?

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b. What is the maximum velocity before the box slips off?

| |

c. What is the acceleration of the 200-kg merry-go-round (β = ½) exerting by a 50-N force along the outer rim?

| |

d. How much time will it take to reach the maximum velocity before the box slips off of the merry-go-round?

| |

e. Would this time increase or decrease if μ = 1.0?

| |

32. A 5-m, 75-kg plank is extended 2 m over the edge of a building. What is the maximum distance that a 25-kg child walks out from the building's edge without tipping the plank?

| |

33. Consider the diagram of the printing press on a table.

Determine

|F1 | |

|F2 | |

34. A plank is placed on two scales, which are then zeroed. A 172-cm-tall student lies on the plank resulting in the reading shown.

[pic]

a. What is the student's mass?

| |

b. What is the distance from her feet to her center-of-mass?

| |

35. A 2200-kg trailer is attached to a stationary truck.

[pic]

Determine the normal force at A and B

|A | |

|B | |

36. A 200-N sign hangs from the end of a 5-m pole, which is held at a 37o angle by a horizontal guy wire.

guy wire

pole

37o

Determine the tension in the guy wire.

| |

C. Conservation Laws—Rotation

37. A hoop, cylinder and sphere roll down a 1-m ramp inclined 30o at the same time that a box slides down a frictionless ramp that is also 1 m long and inclined 30o.

a. Derive a formula for determining the velocity of each object when it reaches the bottom of the ramp.

| |

b. What are the velocities when they reach the bottom?

|Hoop (β = 1) | |

|Cylinder (β = 1/2) | |

|Sphere (β = 2/5) | |

|Box (β = 0) | |

c. What is the order in which they reach the bottom?

| |

38. Determine the velocity of a Yo-Yo (β = ½) that "rolls" straight down its string a distance of 0.50 m.

| |

39. A marble (β = 2/5) rolls from rest down a ramp and around a loop (radius = 10 m). Determine

A

B

H 10 m

a. the minimum velocity at B.

| |

b. the minimum height H at A.

| |

40. A string is attached to a 1.0-kg block and is wrapped round a pulley (β = ½, m = 2.0 kg). The block is released from rest and accelerates downward while the pulley rotates.

What is the block's velocity after descending 1 m?

| |

41. Two weights (m1 = 0.60 kg, m2 = 0.40 kg) are connected by a cord that hangs from a pulley (β = ½, M = 0.50 kg).

M

m1

1 m

m2

What is the velocity of m1 after descending 1 m?

| |

42. A string attached to a 20-kg block resting on a table passes over a pulley (β = ½, m = 4 kg) and attaches to a 14-kg mass hanging over the edge of the table. The 20-kg box slide along the table (μ = 0.25) while the 14-kg mass descends 1 m.

20 kg

1 m

14 kg

What is the hanging mass' velocity after descending 1 m?

| |

43. What is the angular momentum of a 0.2-kg ball traveling at 9 m/s on the end of a string in a circle of radius 1 m?

| |

44. What is the angular momentum of Earth, m = 6.0 x 1024 kg?

a. about its axis of rotation (β = 2/5, rplanet = 6.4 x 106 m)

| |

b. in its orbit around the Sun (β = 1, rorbit = 1.5 x 1011 m)

| |

45. Halley's comet follows an elliptical orbit, where its closest approach to the sun is 8.9 x 1010 m and its farthest distance is 5.3 x 1012 m. How many times faster does the comet travel at its fastest compared to its slowest?

| |

46. A child (m = 42 kg) runs toward a stationary merry-go-round (β = ½, m = 180 kg, r = 1.2 m) along a tangent at 3 m/s. The child jumps on the merry-go-round and sets it rotating.

3 m/s

42 kg β = ½

180 kg

1.2 m

What is the speed of the merry-go-round after the child jumps on?

| |

47. The rim of a disk (β = ½, m = M, r = R) rotates at a velocity, V. A ring (β = 1, m = M, r = R) is dropped on top of the disk.

a. Calculate Ltotal before the ring is dropped on the disk.

| |

b. Calculate the velocity after the ring is dropped.

| |

48. Tarzan (100 kg) is on a ledge that is 20 m above Jane (45 kg), who is trapped on a lower ledge. Tarzan grabs a long vine and swings down from the ledge and grabs Jane, who is stationary. The two swing over to a rock ledge on the other side of the river gorge that is 10 m higher than the rock ledge where Jane is trapped. Assuming the vine is long enough, can Tarzan and Jane reach the other side?

T

J

a. Calculate Tarzan's velocity when he grabs Jane.

| |

b. Calculate the velocity after Tarzan grabs Jane.

| |

c. Calculate how high Tarzan swings to the other side.

| |

|d. Did Tarzan and Jane make it? | |

e. What could Tarzan have done to save Jane?

| |

f. How high would Tarzan have to start to save Jane?

| |

g. What minimum initial velocity would Tarzan need to save Jane starting from the original ledge?

| |

49. A 1-kg, disk (β = ½) is placed on a 2-m ramp where the top is 1 m above the base of the ramp. The disk is placed at the top and rolls down to the base of the ramp.

a. What is the disk's velocity when reaches the base?

| |

b. How much time does it take the disk to travel the 2 m?

| |

c. Predict how the following alterations would change the disk's velocity at and time to reach the base of ramp?

|Alteration | Final Velocity |Time |

|A 2.0-kg disk is used | | |

|A 1.0-kg ring (β = 1) is used | | |

|A 3-m ramp is used, but h = 1 m | | |

50. A string attached to a 10-kg box resting on a table passes over a pulley (β = ½, m = 1 kg) and attaches to a 5-kg mass hanging over the edge of the table. The 10-kg box slide 1 m along the table (μ = 0.3) while the 5-kg mass descends.

1 m

a. How much kinetic energy does the system have at the point where the 5-kg mass has descended 1 m?

| |

b. What is the maximum velocity of the system?

| |

51. Halley's Comet has a velocity of 3.88 x 104 m/s when it is 8.9 x 1010 m from the sun. How fast is it traveling when it is 5.3 x 1012 m from the sun?

| |

52. What is the angular momentum of the Moon?

(m = 7.35 x 1022 kg, rmoon = 1.74 x 106 m, rorbit = 3.84 x 108 m, Torbit = Trotation = 2.42 x 106 s)

a. about its axis of rotation (β = 2/5)

| |

b. in its orbit around the Earth (β = 1)

| |

53. A student (m = 75 kg) runs at 5 m/s tangentially toward a merry-go-round (β = ½, m = 150 kg, r = 2 m) rotating at 2 m/s, jumps on the merry-go-round and sets it rotating. What is the velocity of the student after he jumps on to the merry-go-round?

| |

54. A 2-kg block and a 1-kg sphere hang from 2-m strings. The sphere is raised to a horizontal position and swings toward the block and collides with it.

1 kg

2 kg

a. What is the sphere's velocity before the collision?

| |

Assume that the collision is inelastic.

b. What is the sphere-block's velocity after the collision?

| |

c. What is the maximum height reached after the collision?

| |

d. What is the maximum height reached after the collision if the block and sphere exchange positions initially?

| |

The sphere is raised to a horizontal position initially and then collides elastically with the block.

e. What are the velocities after the collision?

| |

f. What are the maximum heights reached by the block and sphere?

| |

g. Was potential energy conserved after the collision?

| |

D. Simple Harmonic Motion (SHM)

Questions 55-58 A spring bob in SHM has amplitude A and period T.

55. What is the total distance traveled by the bob after time T?

(A) 0 (B) ½A (C) 2A (D) 4A

56. What is the total displacement after time T?

(A) 0 (B) ½A (C) 2A (D) 4A

57. How long does it take the bob to travel a distance of 6A?

(A) ½T (B) ¾T (C) 5/4T (D) 3/2T

58. At what point in the motion is v = 0 and a = 0 simultaneously?

(A) x = 0 (B) 0 < x < A(C) x = A (D) no point

59. A mass on the end of a spring oscillates in simple harmonic motion with amplitude A. If the mass doubles but the amplitude is not changed, what happen to the total energy?

(A) decrease (B) no change (C) increase

60. If the amplitude of a simple harmonic oscillator is doubled, which quantity will change the most?

(A) T (B) v (C) a (D) K + U

61. A spring with mass m has period T. If m is doubled, what is the new T?

(A) T/√2 (B) T (C) √2T (D) 2T

Questions 62-63 Consider the periods of pendulums A and B,

62. Which period is greater when LA = LB, but mA > mB?

(A) A (B) B (C) tie

63. Which period is greater when mA = mB, but LA > LB?

(A) A (B) B (C) tie

64. A grandfather clock has a weight at the bottom of the pendulum that can be moved up or down. If the clock is running slow, should the weight be moved up or down?

(A) up (B) down (C) neither will work

65-66 Consider the following options.

(A) add mass to the oscillator

(B) move the oscillator to an elevator accelerating down

(C) move the oscillator to an elevator accelerating up

(D) move the oscillator to the Moon

65. Which will decrease the period of a pendulum?

66. Which will change the period of oscillation on a spring?

67. After a pendulum starts swinging, its amplitude gradually decreases with time because of friction. What happens to the period of the pendulum during this time

(A) decreases (B) no change (C) increases

68. When you sit on a swing, the period of oscillation is T1. When you stand on the same swing, the period of oscillation is T2. Which is true?

(A) T1 < T2 (B) T1 = T2 (C) T1 > T2

69. When a 50 kg person sits on a swing, the period of oscillation is T1, when a 100 kg person sits on the same swing, the period of oscillation is T2. Which is true?

(A) T1 < T2 (B) T1 = T2 (C) T1 > T2

70. Consider the graph

of one cycle of SHM.

a. Determine the time in terms of T for each situation.

| |Maximum up |Zero |Maximum down |

|Acceleration | | | |

|Velocity | | | |

b. Determine the following when m = 1 kg, k = 100 N/m and A = 0.1 m.

|amax | |

|vmax | |

|T | |

|Kmax | |

|Umax | |

c. Graph the potential energy (----), kinetic energy (•••) and total energy (––) for one complete oscillation.

|0.5 J | | | | |

| | | | | |

|0 J | | | | |

| |¼T |¾T |

d. complete the following chart (x = +A at t = 0 s)

|t |¼T |½T |¾T |1T |

|x | | | | |

|v | | | | |

|a | | | | |

|F | | | | |

e. How do the following change if the amplitude is 0.2 m?

|Max acceleration |Max velocity |Period |

| | | |

71. A 1-kg ball on the end of a 1-m string is set in motion by pulling the ball out so that it is raised 0.015 m. Determine

a. the maximum speed

| |

b. the period of oscillation.

| |

c. What would the period be with the following changes?

|m = 4 kg |L = 4 m |g = 40 m/s2 |

| | | |

72. Consider the diagram of one cycle of SHM.

[pic]

a. Determine the time (in terms of T) for each situation.

| |Maximum up |Zero |Maximum down |

|Acceleration | | | |

|Velocity | | | |

b. Determine the following when m = 1 kg, k = 100 N/m and A = 0.25 m.

|amax | |

|vmax | |

|T | |

|Kmax | |

|Umax | |

c. Graph the potential energy, kinetic energy and total energy for one complete oscillation.

|3 J | | | | |

| | | | | |

|0 J | | | | |

| |¼T |¾T |

d. complete the following chart (x = 0 at t = 0 s)

|t |¼T |½T |¾T |1T |

|x | | | | |

|v | | | | |

|a | | | | |

|F | | | | |

e. Which is the same when the amplitude is increased to 0.50 m?

|maximum acceleration |maximum velocity |period |

73. A 1-kg ball swings from the ceiling on the end of a 2-m string. The ball starts its swing from a position that is

0.2 m above its lowest point.

a. What is the maximum speed of the ball?

| |

b. What is the period of oscillation for the pendulum?

| |

Practice Multiple Choice

Briefly explain why the answer is correct in the space provided.

1. In the diagram, a car travels clockwise at constant speed.

[pic]

Which letters represent the directions of the car's velocity, v, and acceleration, a?

v a v a

(A) A C (B) C B

(C) C A (D) D A

| | |

2. A racing car is moving around the circular track of radius 300 m. At the instant when the car's velocity is directed due east, its acceleration is 3 m/s2 directed due south. When viewed from above, the car is moving

(A) clockwise at 30 m/s (B) counterclockwise at 30 m/s

(C) clockwise at 10 m/s d. counterclockwise at 10 m/s

| | |

3. A person weighing 800 N on earth travels to another planet with the same mass as earth, but twice the radius. The person's weight on this other planet is most nearly

(A) 200 N (B) 400 N (C) 800 N (D) 1600 N

| | |

4. The disk is rotating counterclockwise when the ball is projected outward at the instant the disk is in the position shown.

Which of the following best shows the subsequent direction of the ball relative to the ground?

(A) ∧ (B) ⇐ (C) ⋄ (D) ⇒

| | |

5. A ball is released from rest at position P swings through position Q then to position R where the string is again horizontal.

What are the directions of the ball's acceleration at positions, Q and R?

Q R Q R

(A) ⇔ ⇐ (B) ⇔ ⋄

(C) ⇐ ⇔ (D) ⇐ ⇓

| | |

6. A 5-kg sphere is connected to a 10-kg sphere by a rod.

[pic]

The center of mass is closest to

(A) A (B) B (C) C (D) D

| | |

7. A ball attached to a string is moved at constant speed in a horizontal circular path. A target is located near the path of the ball as shown in the diagram.

[pic]

At which point along the ball's path should the string be released, if the ball is to hit the target?

(A) A (B) B (C) C (D) D

| | |

Questions 8-9 refer to a ball that is tossed straight up from the surface of a small asteroid with no atmosphere. The ball rises to a height equal to the asteroid's radius and then falls straight down toward the surface of the asteroid.

8. What forces act on the ball while it is on the way up?

(A) a decreasing gravitational force that acts downward

(B) an increasing gravitational force that acts downward

(C) a constant gravitational force that acts downward

(D) a constant gravitational force that acts downward and a decreasing force that acts upward

| | |

9. The acceleration of the ball at the top of its path is

(A) at its maximum value for the ball's flight

(B) equal to the acceleration at the surface

(C) equal to one-half the acceleration at the surface

(D) equal to one-fourth the acceleration at the surface

| | |

10. The diagram shows a 5.0-kg bucket of water being swung in a horizontal circle of 0.70-m radius at a constant speed of 2.0 m/s.

The centripetal force on the bucket of water is

(A) 5.7 N (B) 29 N (C) 14 N (D) 200 N

| | |

Questions 11-12 A 125-N board is 4 m long and is supported by vertical chains at each end. A person weighing 500 N is sitting on the board. The tension in the right chain is 250 N.

11. What is the tension in the left chain?

(A) 250 N (B) 375 N (C) 500 N (D) 625 N

| | |

12. How far from the left end of the board is the person sitting?

(A) 0.4 m (B) 1.5 m (C) 2 m (D) 2.5 m

| | |

13. A square piece of plywood on a horizontal tabletop is subjected to the two horizontal forces shown above.

[pic]

Where should a third force of magnitude 5 N be applied to put the piece of plywood into equilibrium?

| A B |

|C |

| | | | |

| D | | | |

| |

| | |

14. A 5-m, 100-kg plank rests on a ledge with 2 m extended out.

[pic]

How far can a 50-kg person walk out on the plank past the edge of the building before the plank just begins to tip?

(A) ½ m (B) 1 m (C) 3/2 m (D) 2 m

| | |

15. The radius of the earth is approximately 6,000 km. The acceleration of an astronaut in a perfectly circular orbit 6,000 km above the earth would be most nearly

(A) 0 m/s2 (B) 2.5 m/s2 (C) 5 m/s2 (D) 10 m/s2

| | |

16. The diagram represents two satellites of equal mass, A and B, in circular orbits around a planet.

[pic]

Comparing the gravitational force between satellite and planet, B's gravitational force compared to A's is

(A) half as great (B) twice as great

(C) one-fourth as great (D) four times as great

| | |

17. The system is balanced when hanging by the rope.

[pic]

What is the mass of the fish?

(A) 1.5 kg (B) 2 kg (C) 3 kg (D) 6 kg

| | |

18. A ball attached to a string is whirled around in a horizontal circle with radius r, speed v and tension T. If the radius is increased to 4r and the tension remains the same, then the speed of the ball is

(A) ¼v (B) ½v (C) v (D) 2v

| | |

19. A 0.4-kg object swings on the end of a string. At the bottom of the swing, the tension in the string is 6 N. What is the centripetal force acting on the object at the bottom of the swing?

(A) 2 N (B) 4 N (C) 6 N (D) 10 N

| | |

20. Two wheels, fixed to each other, are free to rotate about a frictionless axis perpendicular to the page. Four forces are exerted tangentially to the rims of the wheels.

[pic]

The net torque on the system about the axis is

(A) zero (B) FR (C) 2FR (D) 5FR

| | |

21. Mars has a mass 1/10 that of Earth and a diameter 1/2 that of Earth. The acceleration of a falling body near the surface of Mars is most nearly

(A) g/5 (B) 2g/5 (C) g/2 (D) g

| | |

22. A satellite of mass m and speed v moves in a stable, circular orbit around a planet of mass M. What is the radius of the satellite’s orbit?

(A) GM/mv (B) Gv/mM (C) GM/v2 (D) GmM/v

| | |

23. A wheel of radius R is mounted on an axle so that the wheel is in a vertical plane. Three small objects having masses m, M, and 2M, respectively, are mounted on the rim.

[pic]

What is m in terms of M when the wheel is stationary?

(A) 1/2 M (B) M (C) 3/2 M (D) 2 M

| | |

24. In each case the unknown mass m is balanced by a known mass M1 or M2.

[pic]

What is the value of m in terms of the known masses?

(A) M1 + M2 (B) (M1 + M2)/2

(C) M1M2 (D) (M1M2)½

| | |

25. An asteroid moves in an elliptic orbit with the Sun at one focus.

Which of the following increases as the asteroid moves from point P in its orbit to point Q?

(A) Speed (B) Angular momentum

(C) Total energy (D) Potential energy

| | |

26. A satellite S is in an elliptical orbit around a planet P with r1 and r2 being its closest and farthest distances, respectively, from the center of the planet. If the satellite has a speed v1 at its closest distance, what is its speed at its farthest distance?

(A) (r1/r2)v1 (B) (r2/r1)v1

(C) (r2 – r1)v1 (D) ½(r1 + r2)v1

| | |

27. A satellite of mass M moves in a circular orbit of radius R at a constant speed v. Which must be true?

I. The net force on the satellite is equal to Mv2/R and is directed toward the center of the orbit.

II. The net work done on the satellite by gravity in one revolution is zero.

III. The angular momentum of the satellite is a constant.

(A) I only (B) III only (C) I and II (D) I, II, and III

| | |

28. For which motions is there a variable force involved?

(A) Constant speed in a straight line

(B) Simple harmonic motion

(C) Constant speed in a circle

(D) Constant acceleration in a straight line

| | |

Questions 29-30 A sphere of mass M, radius R, and β = 2/5, is released from rest at the top of an inclined plane of height h.

[pic]

29. If the plane is frictionless, what is the speed of the center of mass of the sphere at the bottom of the incline?

(A) (2gh)½ (B) 2Mgh (C) 2MghR2 (D) 5gh

| | |

30. If the plane has friction so that the sphere rolls without slipping, what is the speed at the bottom of the incline?

(A) (2gh)½ (B) 2Mgh (C) 2MghR2 (D) (10gh/7)½

| | |

31. A particle of mass, m, moves with a constant speed v along the dashed line y = a.

When the x-coordinate of the particle is xo, the magnitude of the angular momentum of the particle with respect to the origin of the system is

(A) zero (B) amv (C) xomv (D) (vo2 + a2)½mv

| | |

Questions 32-36 A block oscillates without friction on the end of a spring. The minimum and maximum lengths of the spring as it oscillates are, respectively, xmin and xmax.

[pic]

The graphs below can represent quantities associated with the oscillation as functions of the length x of the spring.

(A) (B)

(C) (D)

32. Which graph can best represent the total mechanical energy of the block-spring system as a function of x?

| | |

33. Which graph can best represent the kinetic energy of the block as a function of x?

| | |

34. Which graph can best represent the potential energy of the block as a function of x?

| | |

35. Which graph can best represent the acceleration of the block as a function of x?

| | |

36. Which graph can best represent the velocity of the block as a function of x?

| | |

37. A block attached to the lower end of a vertical spring oscillates up and down. The period of oscillation depends on which of the following?

I. Mass of the block

II. Amplitude of the oscillation

III. Spring constant

(A) I only (B) II only (C) III only (D) I and III only

| | |

38. When a 1-kg bob is attached to a spring, the period of oscillation is 2 s. What is the period of oscillation when a 2-kg bob is attached to the same spring?

(A) 0.5 s (B) 1.0 s (C) 1.4 s (D) 2.8 s

| | |

39. A pendulum and a mass hanging on a spring both have a period of 1 s on Earth. They are taken to planet X, which has twice the gravitational acceleration g as Earth. Which is true about the periods of the two objects on planet X compared to their periods on Earth?

(A) Both are shorter.

(B) Both are longer.

(C) The pendulum is longer and the spring is the same.

(D) The pendulum is shorter and the spring is the same.

| | |

40. The graph is of the displacement x versus time t for a particle in simple harmonic motion with a period of 4 s.

[pic]

Which graph shows the potential energy of the particle as a function of time t for one cycle of motion?

(A) (B) (C) (D)

| | |

41. Two identical springs are hung from a horizontal support. When a 1.2-kg block is suspended from the pair of springs, each spring is stretched an additional 0.15 m.

The spring constant of each spring is most nearly

(A) 40 N/m (B) 48 N/m (C) 60 N/m (D) 80 N/m

| | |

42. A ball is dropped from a height of 10 m onto a hard surface so that the collision at the surface may be assumed elastic. Under such conditions the motion of the ball is

(A) simple harmonic with a period of about 1.4 s

(B) simple harmonic with a period of about 2.8 s

(C) simple harmonic with an amplitude of 5 m

(D) periodic but not simple harmonic

| | |

43. An object swings on the end of a cord as a simple pendulum with period T. Another object oscillates up and down on the end of a vertical spring, also with period T. If the masses of both objects are doubled, what are the new values for the periods?

Pendulum Spring Pendulum Spring

(A) T/√2 √2T (B) T √2T

(C) T T (D) √2T T

| | |

44. When a mass m is hung on a spring, the spring stretches a distance d. If the mass is then set oscillating on the spring, the period of oscillation is proportional to

(A) (d/g)½ (B) (g/d)½ (C) (d/mg)½ (D) (m2g/d)½

| | |

45. A 3-kg block is hung from a spring, causing it to stretch 12 cm at equilibrium. The 3-kg block is then replaced by a 4-kg block, and the new block is released from the spring when it is unstretched. How far will the 4-kg block fall before its direction is reversed?

(A) 9 cm (B) 18 cm (C) 24 cm (D) 32 cm

| | |

46. A spring is fixed to the wall at one end. A block of mass M attached to the other end of the spring oscillates with amplitude A on a frictionless, horizontal surface. The maximum speed of the block is v.

[pic]

The spring constant is

(A) Mg/A (B) Mgv/2A (C) Mv2/2A (D) M(v/A)2

| | |

47. A sphere of mass m1 is attached to a spring. A second sphere of mass m2 is suspended from a string of length L, If both spheres have the same period of oscillation, which of the following is an expression for the spring constant?

(A) L/m1g (B) g/m2L (C) m1L/g (D) m1g/L

| | |

Practice Free Response

1. A roller coaster ride at an amusement park lifts a car of mass 700 kg to point A at a height of 90 m above the lowest point on the track, as shown above. The car starts from rest at point A, rolls with negligible friction down the incline and follows the track around a loop of radius 20 m. Point B, the highest point on the loop, is at a height of 50 m above the lowest point on the track.

P

a. (1) Indicate on the figure the point P at which the maximum speed of the car is attained.

(2) Calculate the value vmax of this maximum speed.

| |

b. Calculate the speed vB of the car at point B.

| |

c. (1) Draw and label vectors to represent the forces acting on the car when it is upside down at point B.

| |

| |

(2) Calculate all the forces identified in (c1).

| |

2. A string attached to a 20-kg block resting on a table passes over a pulley (β = ½, m = 10 kg) and attaches to a 10-kg mass hanging over the edge of the table. The 20-kg box slide along the table (μ = 0.30) while the 10-kg mass descends 2 m.

2 m

Determine the

a. force of friction on the 20-kg block as it slides.

| |

b. force of gravity on the 10 kg mass.

| |

c. net force rotating the pulley.

| |

d. acceleration of the pulley at the rim.

| |

e. velocity when the system has moved 2 m.

| |

f. loss of potential energy as the 10-kg mass falls 2 m.

| |

g. work done by friction as the 20-kg block slides 2 m.

| |

h. velocity of the system

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3. A 0.5-kg hoop (β = 1) rolls from rest at the top of the ramp of length L = 2 m and angle θ = 30o. The table height H = 1 m.

a. Determine the potential energy of the hoop at the top of the ramp, where Ug = 0 at the floor.

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b. The hoop rolls down the ramp and then onto the floor. Determine the hoop's

(1) speed at the bottom of the ramp.

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(2) speed just before it hits the floor.

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(3) translational kinetic energy before it hits the floor.

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(4) percentage of total energy that is rotational kinetic energy just before it hits the floor.

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c. The hoop is replaced by a disk (β = ½). Comparing the hoop (β = 1) with the disk (β = ½) just before they lands on the floor, which would

(1) have the greatest % rotational kinetic energy?

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(2) land furthest from the base of the table?

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(3) have the most kinetic energy just before it landed?

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4. The graph shows a system in simple harmonic motion.

[pic]

Complete the chart with either +, 0, or –.

|t |0 s |1 s |2 s |3 s |

|x (m) | | | | |

|v (m/s) | | | | |

|a (m/s2) | | | | |

|F (N) | | | | |

|U (J) | | | | |

|K (J) | | | | |

5. A 3.0 kg bob swings on the end of a 1.0 m string. The potential energy U of the object as a function of distance x from its equilibrium position is shown. This particular object has a total energy E of 0.4 J.

[pic]

a. What is the greatest distance x for the pendulum bob?

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b. How much time does it take the pendulum to go from the greatest +x to the greatest –x?

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c. Determine the bob's kinetic energy when x = -7 cm.

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d. What is the object's speed at x = 0?

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6. The cart of mass m with four wheels each of mass m/4 and β = ½ is released from rest and rolls from height h. After rolling down the ramp and across the horizontal surface, the cart collides and sticks with a bumper of mass 3m attached to a spring, which has a spring constant k.

[pic]Given: m = 1 kg, h = 0.50 m, k = 250 N/m, determine

a. the potential energy of the cart at the top of the ramp.

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b. the speed of the cart at the bottom of the ramp.

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c. the carts translational kinetic energy before the collision.

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d. the speed of the cart just after the collision.

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e. the translational kinetic energy of the cart and bumper just after the collision.

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f. the amount that the spring is compressed.

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Physics

is Phun

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