EXAM I, PHYSICS 1306 - Texas Tech Physics & Astronomy



FINAL EXAM, PHYSICS 1408, August 5, 2010, Dr. Charles W. Myles

INSTRUCTIONS: Please read ALL of these before doing anything else!!!

1. PLEASE put your name on each sheet of paper & write on 1 side of the paper only!! This wastes paper, but makes my grading easier! PLEASE DON’T write on the exam sheets, there isn’t room!

2. PLEASE show all work, writing the essential steps in the solutions. Write formulas first, then put in numbers. Partial credit will be LIBERAL, provided that essential work is shown. (If you don’t show your work, how will I be able to give you partial credit?) Organized, logical, easy to follow work will receive more credit than disorganized work.

3. The setup (PHYSICS) of a problem counts more heavily than the math of working it out.

4. PLEASE write neatly. PLEASE: Before handing in your solutions, a) number the pages & put the pages in numerical order, b) put the problem solutions in numerical order, & c) clearly mark your final answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.

5. The words “EXPLAIN”, “DISCUSS”, “DEFINE” below mean to answer mostly in ENGLISH, NOT symbols! Answers to such questions containing ONLY symbols will get NO CREDIT!

NOTE: I HAVE 49 EXAMS TO GRADE!!! PLEASE HELP ME GRADE THEM EFFICIENTLY BY FOLLOWING THESE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A LOWER GRADE!! THANK YOU!!

Up to three (3) 8.5’’ x 11’’ pieces of paper with anything written on them & a calculator are allowed. Problem 1 (Conceptual) AND Problem 2 (Rotations) ARE REQUIRED! Work EITHER Problem 3 (Momentum) OR Problem 4 (Momentum). Choose TWO (2) of the others for five (5) problems total. Each is equally weighted & worth 20 points, for a total of 100 points on this exam. NOTE: Some answers to the following problems are very large or very small numbers! PLEASE express such answers in scientific (power of 10) notation! Thanks!

1. MANDATORY Conceptual Questions!!! Answer briefly. Most answers should be complete,

grammatically correct English sentences. Keep formulas to a minimum. Use WORDS instead! If you use a formula, YOU MUST DEFINE EVERY SYMBOL you use. (Note: Answers giving ONLY symbols, with no explanation about what they mean, will receive NO credit! PLEASE answer all questions!). Parts a – j are equally weighted & worth 2 points each.

a. State Newton’s 1st Law of Motion. How many masses at a time does it apply to?

b. State Newton’s 2nd Law of Motion. How many masses at a time does it apply to?

c. State Newton’s 3rd Law of Motion. How many masses at a time does it apply to?

d. State Newton’s Universal Law of Gravitation.

e. State the Work-Energy Principle.

f. When discussing the Work-Energy Principle, I stated that it is an already familiar Law expressed in

work-energy language. What Law was I referring to?

g. State the Principle of Conservation of Mechanical Energy. Which kinds of forces are required to be

present in order for this principle to hold?

h. State the Principle of Conservation of Momentum. Under what conditions is momentum conserved?

i. State Newton’s 2nd Law in terms of Momentum. (Hint: State Newton’s original form for his 2nd Law! ∑F = ma will get ZERO credit!).

j. State Newton’s 2nd Law for Rotational Motion. (∑F = ma will get ZERO credit!).

2. MANDATORY Rotational Motion Problem!!! In our discussion of rotational motion of rigid objects, I stressed that Ch. 10 is about Newton’s Laws applied to rotating objects, expressed in Rotational Language instead of Translational Language. I also stressed that every rotational motion concept or relation has a completely analogous translational motion concept or relation that we already know. Using these ideas, plus attending class yesterday, should enable you to solve the following: The figure shows a rigid, solid SPHERE of radius R = 1.8 m. The sphere’s mass is M = 4.2 kg. It’s moment of inertia is I = (2/5)M(R)2. (Note: So, it is ISN’T a uniform disk, & the disk moment of inertia, Idisk = (½)MR2 should NOT be used!!) At time t = 0, it starts from rest (initial angular velocity ω0 = 0) & begins to rotate counterclockwise about an axis passing through center of the sphere & perpendicular to the page. The figure view looks down at the rotation plane, with rotation in the counter-clockwise direction, as indicated. The sphere’s angular acceleration is α = 0.4 rad/s2.

CONTINUED ON NEXT PAGE!!

PROBLEM 2 CONTINUED!!

a. Calculate it’s angular velocity ω & angular displacement θ after it has been rotating for time, t = 22 s.

Parts b & c are about a point P on the equator, a distance R = 1.8 m from the rotation axis, as shown.

b. Calculate the velocity v (m/s) of point P at the time t = 22 s. v is a vector tangent to the rim. Calculate the centripetal acceleration aR (m/s2) of P at that same time. What is the vector direction of aR?

c. Calculate the tangential acceleration atan (m/s2) of P at t = 22 s. atan is a vector tangent to the equator.

As I stressed in our brief discussion, the angular acceleration α = 0.4 rad/s2 of the sphere must be caused by a torque τ applied to it (torque is the rotational analogue of force).

d. Use Newton’s 2nd Law for Rotational Motion along with the angular acceleration α = 0.4 rad/s2 and the moment of inertia I = (2/5)M(R)2 to calculate the net torque τ which causes the sphere to rotate.

e. Assume that the force F which causes the torque just calculated is a vector applied tangent to the sphere’s surface, at it’s equator at R = 1.8 m, as in the figure. Use the relation between forces & torques to calculate the force F responsible for giving the sphere the torque τ.

NOTE: EITHER PROBLEM 3 (Momentum) OR PROBLEM 4 (Momentum) IS REQUIRED!!!!!

Don’t forget to look at the BONUS questions (# 8 on the last page)!

3. See figure. A bullet, mass m = 0.06 kg, traveling at velocity v strikes & becomes embedded in a block of wood, mass M = 4.1 kg, initially at rest on a horizontal surface. The block-bullet combination then moves to the right. After the collision, their velocity is V = 7.5 m/s.

a. Calculate the momentum & the kinetic energy of the bullet-block combination just after the collision.

b. Calculate the momentum of the bullet just before the collision. Calculate its velocity v just before the collision. What Physical Principle did you use to answer this?

c. Calculate the bullet kinetic energy just before the collision. Was kinetic energy conserved in the collision? Explain (with brief, complete, grammatically correct English sentences! Hint: Please THINK before answering! Compare the kinetic energy found here with that found in part a! It’s Physically Impossible to GAIN kinetic energy in a collision!)

d. Calculate the impulse Δp delivered to the block by the bullet. Stated another way, calculate the

momentum change of the block in the collision.

e. If the collision time was Δt = 2.5 ( 10-3 s, calculate the average force exerted by the bullet on the block. What Physical Principle did you use to answer this question?

NOTE: EITHER PROBLEM 3 (Momentum) OR PROBLEM 4 (Momentum) IS REQUIRED!!!!!

Don’t forget to look at the BONUS questions (# 8 on the last page)!

4. See figures. Two balls undergo an elastic collision as they approach each

other head-on. The “before” figure shows them before the collision & the

“after” figure shows them afterward. The masses are m1 = 0.42 kg,

m2 = 0.27 kg. The initial velocity of m1 is v1 = 4.6 m/s & that of m2

is v2 = -3.4 m/s. The initial velocities are in opposite directions, as in the

“before” figure. After the collision, their velocities v1´ & v2´ are again in

opposite directions, as in the “after” figure.

a. Calculate the total momentum p1 + p2 of the two balls before the collision. (Hint: Don’t forget that momentum and velocity are vectors & DIRECTION matters!). Calculate the total kinetic energy KE1 + KE2 of the two balls before the collision.

b. Calculate the total momentum p1´+ p2´ of the two balls after the collision. Calculate the total kinetic energy KE1´ + KE2´of the two balls after the collision. (Hint: Both can be calculated WITHOUT first finding the velocities v1´ & v2´ of the balls after the collision!). What physical principles did you use to find these results? Is kinetic energy conserved in this collision?

c. Calculate the velocities v1´ & v2´ of the balls after the collision.

d. Calculate the impulse Δp2 experienced by m2 due to it’s collision with m1. Stated another way, calculate the momentum change of m2 in the collision.

e. If the collision time was Δt = 3 ( 10-3 s, use the calculated impulse to calculate the average force exerted by m1 on m2. What physical principle did you use to calculate this force?

NOTE: WORK ANY TWO (2) OF PROBLEMS 5, 6, or 7!!!!!

Don’t forget to look at the BONUS questions (# 8 on the last page)!

5. See figure. A mass m = 7.4 kg slides with initial velocity v = 4.2 m/s (to the left in the figure) across a horizontal, frictionless surface until it encounters a spring with constant k = 275 N/m. It comes instantaneously to rest after compressing the spring a distance x. (Hint: In the following, PLEASE try to take square roots properly!)

a. Calculate the initial kinetic energy of the mass (left hand figure).

Note that the gravitational potential energy of the mass is unchanged for parts b & c, so it is irrelevant to both of those parts!

b. Calculate the elastic potential energy of the spring-mass system at the point where the mass is instantaneously at rest (right hand figure). Also, calculate the distance x the spring is compressed there. What physical principle did you use to find these?

c. Calculate the elastic (spring) potential energy of the spring-mass system & the distance the spring has been compressed when the mass’s speed has slowed to 1.8 m/s. (This isn’t shown in the figures! This occurs after the mass touches the spring & before it has come to rest as in the right hand figure! Note: Answers obtained by setting the kinetic energy equal to the potential energy at this point will get ZERO credit! At this stage of your physics learning, such answers show a serious lack of understanding of the Principle of Mechanical Energy Conservation!)

d. Note that the elastic (spring) potential energy of the mass is unchanged for the following question, so it is irrelevant to it! The mass in the left hand figure was given its initial velocity V = 4.2 m/s by sliding it from rest down a frictionless inclined plane from a height h. (This isn’t shown in the figures! It happened sometime before the situation shown in the left hand figure!) Calculate the potential energy the mass had at the top of the inclined plane & the height h from which the mass started. What physical principle did you use to find these results?

e. Calculate the FORCE (magnitude & direction) the spring exerts on the mass at the final position (in right hand figure) where the mass has stopped moving.

NOTE: WORK ANY TWO (2) OF PROBLEMS 5, 6, or 7!!!!!

Don’t forget to look at the BONUS questions (# 8 on the last page)!

6. See figure. A box of mass m = 25 kg is accelerated by being pushed across a flat

table, using a force F = 50 N, which makes an angle θ = 30° below the horizontal,

as shown. The coefficient of kinetic friction between the box & the surface is

μk = 0.14. There is no vertical acceleration.

a. Sketch the free body diagram for the box, properly labeling all forces. (Note: If you don’t make a sketch, you will lose points!). Calculate the horizontal & the vertical components of the force F.

b. Calculate the weight of the box and use Newton’s 2nd Law in the vertical direction to calculate the normal force FN between the box & the surface. Also calculate the friction force Ffr on the box as it moves to the right. (Note: Answers stating that FN is equal & oppositely directed to the weight will get ZERO credit! At this stage of your physics learning, such answers show a serious lack of understanding of vector forces & Newton’s 2nd Law!!!!)

c. Use Newton’s 2nd Law to find the acceleration a of the box. What forces cause this acceleration?

d. Calculate the work done by the constant force F, the work done by the friction force Ffr, & the net work done by all forces as the box moves to the right through a horizontal displacement x = 14 m.

e. Assuming that the box starts from rest, use energy methods along with the results of part d to calculate it’s kinetic energy & velocity after it has moved the horizontal displacement x = 14 m of part d. (Solutions obtained using the Ch. 2 kinematic equations will get ZERO credit!)

NOTE: WORK ANY TWO (2) OF PROBLEMS 5, 6, or 7!!!!!

Don’t forget to look at the BONUS questions (# 8 below)!

7. See figure. Two masses (mI = 16 kg & mII = 28 kg) are connected by a massless cord over a massless, frictionless pulley as shown. mI sits on a frictionless table. The masses are released, & mI moves to the right with acceleration a while mII moves down with the same acceleration.

a. Sketch the free body diagrams for the two masses, properly labeling all forces. The tension in the cord is FT. (Note: If you don’t make a sketch, you will lose points!).

The unknowns are the acceleration a & the tension FT. For parts b & c, 2 simultaneous linear equations in 2 unknowns must be used & algebra must be done to solve for a & FT. (FT can’t possibly be equal in magnitude to the weight mg, or a would be zero! Also, a can’t possibly be equal to the gravitational acceleration g downward or FT would be zero!)

b. By applying Newton’s 2nd Law to the two masses, find the two equations needed to solve for a & FT.

More credit will be given if you leave these equations in terms of symbols with no numbers substituted than if you substitute numbers into them. (Note: I don’t mean to just write it abstractly as ∑F = ma. I

mean to write the equations which result when Newton’s 2nd Law is APPLIED to this problem!)

c. Using the equations from part b, calculate a & FT (in any order).

d. Calculate the work done by the force FT, the work done by gravity, & the net work done by all forces

acting on mII as it falls a distance y = 2.5 m.

e. Assuming that mII starts from rest, use energy methods along with the results of part d to calculate it’s

kinetic energy & velocity after it has fallen the distance y = 2.5 m of part d. (Solutions obtained using

the Ch. 2 kinematic equations will get ZERO credit!)

8. BONUS QUESTIONS! (10 bonus points total!) Answer briefly, in a few complete, grammatically correct English sentences. You may supplement these sentences with equations, but keep these to a minimum and EXPLAIN what the symbols mean! I want most of the answer to be in WORDS! (Note: Answers with ONLY symbols, with no explanation about what they mean, will receive NO credit!)

a. See figure. A hockey puck is sliding (to the right) at constant

velocity across a flat, horizontal, frictionless ice surface. Which

of the sketches in the figure is the correct free body diagram for

this puck? WHY? Explain your answer using Newton’s Laws!

(Hint: Is there a net force in the direction of the puck’s motion?)

b. See figure! A child sits in a wagon moving to the right (x-direction) at constant velocity v0x. She throws an apple straight up (from her viewpoint) with an initial velocity v0y while she continues to travel forward at v0x. Neglect air resistance. Will the apple land behind the wagon, in front of the wagon, or in the wagon? WHY? Explain (briefly!) your answer. Use what you know about projectiles!. Make a sketch of the situation to illustrate your explanation. (Note: For full credit, you MUST make a sketch!).

c. See figure. Paul & Kathleen start from rest (on the left) on two different shaped, frictionless water slides. They start at the SAME vertical height h (Note: The figure shows them at different heights because it shows them AFTER they have started down!). Which rider is moving faster at the bottom? What Physical Principle did you use to answer this? If they start at the same time, which rider gets to the bottom first? Why ? (Answer in words!!)

Sometimes this session, I briefly shared with you some interesting (to me!) facts (not in the book!) about Newton’s scientific achievements & also about his life & his dealings with other scientists. If you were in class when I talked about these things, you should be able to answer the following questions. For each, write a few complete sentences.

d. Tell me about ONE of Newton’s scientific achievements OTHER THAN his development of his three Laws of Motion. BE SPECIFIC!

e. Tell me about ONE of the facts I shared with you about Newton’s life & his dealings with other scientists. BE SPECIFIC! [pic]

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( FT

h

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before

m = 7.4 kg

v2

P

[pic]

k = 275 N/m

(

F

v1

mII

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R

h

h

[pic]

ω

(

|

|

h

|

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(

FT (

mI

[pic]

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after

v1´

v2´

v = ?

Initially겘곧곰곹곺과괾굂굋굌굍굎굝굟굢굸궑궰긋긒긗, v = 4.2 m/s Finally, x = ?, v = 0

m = 0.06 kg

↓

↑

M = 4.1 kg

V = 7.5 m/s

Both start at

the same height!

a (

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h

h

( a

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h

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