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UNIVERSITY OF WATERLOO

DEPARTMENT OF PHYSICS

Physics 115 - Mechanics

Final Examination

December 8, 2006 9:00– 11:30 Time: 2.5 Hours

Name: (Print) Division

ID # Signature

Professor: Circle One

R. Jayasundera [Mec. St. 4]008 D. Strickland [Elec. St. 4]005 F. Mansour [Chem]001

R. Jayasundera [Mec. St. 8]007 D. Strickland [Elec. St. 8]006 F. Mansour [Env.Chem] 002

R. Allahverdi [Comp.]004 M. Fich [Civil] 003 M. Balogh [Softw.]009

Aids Permitted: Calculator, attached formula sheet

ATTEMPT ALL QUESTIONS

Important Instructions - Read Carefully

1. Part A has 20 questions, worth 20%.

2. Part B has 6 questions, worth 80%; 1 question per page; recommended average time per question: approximately 20 minutes

3. Check that your booklet has 13 pages (including 1 blank sheet at the back and this cover plus a formula sheet). Questions are to be answered on this paper. Use the space under each question first. If necessary continue on back of previous sheet and then on the blank sheet, and indicate by writing “continued at left” or “continued on page 11”. Answers must include units where appropriate.

4. In Part A, the letter corresponding to each answer must be entered on your computer sheet in dark lead PENCIL only. Avoid erasures by checking answers before marking them. Mark your I.D. Number and Division Number in the space provided. Also (THIS IS IMPORTANT) fill in the correct ovals for your I.D. Number and Division. Do not mark “card number” column. Check that you have not missed, or put double marks in any row in the answer field.

5. Print name, etc., in the upper right part of your computer sheet and at the top of this page.

6. There is no penalty for marking incorrect answers.

7. **NOTE: KEEP COMPUTER SHEET FACE DOWN WHEN NOT IN USE **

8. **NOTE: COMPUTER SHEETS WILL BE COLLECTED AFTER 2 HOURS

9. In Part B, answers must be entered in the boxes provided. These will be hand-marked, and part marks will be given liberally, if your solutions clearly show correct thinking.

10. Ask a proctor to clarify a question if you think that is necessary.

GOOD LUCK !

Use g = 9.80 m/s2

(NOTE: Assume all given data is accurate to three significant figures if needed)

Moment of Inertia (about axes through centre of mass).

Cylinder or Disk (mass M, radius R, axis ( to radius) : ICM = (1/2) MR2. Solid Sphere (mass M, Radius R) : ICM = (2/5) MR2 Hoop (mass M, radius R, axis ( to radius): ICM = MR2 Rod (mass M, length L, axis ( to length) ICM = (1/12) ML2.

|MARKS | | | | | | | |

|ONLY |1B |2B |3B |4B |5B |6B |TOTAL |

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1. Vector [pic] lies in the xy plane. For what orientation of vector [pic] will its components

have opposite signs.

(a) 1st quadrant

(b) 1st and 2nd quadrant

(c) 2nd and 4th quadrants

(d) 1st and 4th quadrants

(e) 2nd and 3rd quadrants

2. A baseball is thrown with an initial speed of [pic]. When it reaches the top of

its trajectory the velocity and acceleration are

(a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

(e) [pic]

3. A crate is sitting in the center of a flatbed truck. If the crate does not slide on the truck,

while the truck is accelerating to the right, the direction of the frictional force acting on

the crate is pointing to the

(a) right

(b) left

(c) no frictional force since the crate is not sliding

(d) all of the above

(e) none of the above

4. The observer in the elevator which is accelerating upward at “a” m/s2, would claim that

the weight of the fish is the

(a) reading of the scale

(b) reading of the scale + ma

(c) reading of the scale – ma

(d) reading of the scale times a

(e) none of the above

(m is the mass of the fish)

5. As a simple pendulum swings back and forth, the forces acting on the suspended masses

are (i) force of gravity (ii) the tension of the cord and (iii) air resistance.

The work done by each of the above forces can be written as Wg, WT, Wair = Wa (note Wg =

work done by gravity etc). Which statement is always true?

(a) Wg > 0. Wa < 0

(b) Wg < 0. WT = 0

(c) Wa = 0, WT = 0

(d) Wa < 0, WT = 0

(e) WT > 0, Wa = 0

6. Two objects are at rest on a frictionless surface. Object 1 has a greater mass than object

2. When a force “F” is applied to object 1, it accelerates through a distance “d”. The

force “F”, is removed from object 1 and applied to object 2. At the moment when object

2 has accelerated through the same distance d, which of the statements is true.

(a) P1 > P2

(b) K2 > K1

(c) K1 = K2

(d) P1 = P2

(e) none of the above

7. A small sports car collides head-on with a massive truck.

Which vehicle experiences the greater magnitude force during the collision?

(a) The car

(b) The truck

(c) The answer depends on the speeds the vehicles were moving

(d) Both experience the same force during the collision

8. Two marbles, one with twice the mass of the other, are dropped to the ground from a

table. Just before hitting the ground, the more massive marble has:

(a) As much kinetic energy as the lighter one.

(b) Twice as much kinetic energy as the lighter one.

(c) Half as much kinetic energy as the lighter one.

(d) Four times as much kinetic energy as the lighter one.

(e) Impossible to determine.

9. Three framed pictures of identical shape and mass m were hanged as shown below.

Which way of hanging the picture requires a stronger string?

(a) 1

(b) 2

(c) 3

(d) 1 and 2 (both are the same)

(e) all are the same

10. A wheel starts from rest and spins with a constant angular acceleration. As time goes on

the acceleration vector for a point on the rim:

(a) decreases in magnitude and becomes more nearly tangent to the rim

(b) decreases in magnitude and becomes more nearly radial

(c) increases in magnitude and becomes more nearly tangent to the rim

(d) increases in magnitude and becomes more nearly radial

(e) increases in magnitude but retains the same angle with the tangent to the rim

11. A ball is rolling freely on a flat surface.

(a) The force of friction points opposite to the motion of the centre of mass.

(b) The force of friction points in the direction of motion of the centre of mass.

(c) The force of friction cannot be determined.

(d) The force of friction is zero.

(e) None of the above.

12. You are holding onto a spring that is already stretched. You then proceed to take that

spring to equilibrium slowly.

(a) The work you do is positive.

(b) The change in potential energy of the spring is positive.

(c) Your work is double that of the work of the spring.

(d) The spring does negative work.

(e) The spring does positive work.

13. You throw a tennis ball of radius R on the floor such that it rolls freely on a flat surface

for some time and then it hits a patch of ice where the coefficient of friction between the

ball and the ice is (s = 0. When the ball hits the patch

(a) It continues to roll as before.

(b) It starts to spin [pic]

(c) It begins to skid [pic]

(d) none of the above

14. A car is driving at a constant speed on the 401 heading towards Montreal. The driver hits

the gas pedal and the engine delivers more power. During the time when the driver is

pressing on the gas pedal (the wheels are always rolling)

(a) The force of friction points opposite to the motion.

(b) The force of friction does not play a role because the speed of the bottom of the wheel is

zero

(c) The force of friction is zero because the wheel is always rolling.

(d) The force of the friction points in the same direction as the velocity of the center of mass.

(e) None of the above

15. An elevator is going up at a constant speed, pulled by a single cable. How does the

tension (T) in the cable compare with the weight of the elevator (W)?

(a) T > W

(b) T = W

(c) T < W

(d) impossible to tell

16. The angular velocity vector of a spinning body points out of the page. If the angular

acceleration vector points into the page then:

(a) the body is slowing down

(b) the body is speeding up

(c) the body is starting to turn in the opposite direction

(d) the axis of rotation is changing orientation

(e) none of the above

17. Which object has the largest moment of inertia? (masses are point masses connected to

massless rods)

18. In which of the following diagrams of a door hinged to a wall at two points A and B at ¼

and ¾ of the length if the door, show the forces in the directions that would result in the

door being in static equilibrium.

19. Which pulley will be spinning fastest when the block hits the ground, if all start from

rest? Assume the strings do not slip and pulleys are discs.

20. The figure shows four disks that are on a frictionless floor. Three forces, of magnitude F,

2F, or 3F, act on each disk, either at the rim, at the center, or halfway between the rim and the center as shown in diagrams. Which disk is in equilibrium?

(a) (b) (c) (d)

1B. A toy gun fires marbles with a velocity v = 7m/s.

(a) To hit a toy soldier that is 1.8 m away and 1.1 m higher than the gun, at what angle

should the gun be aimed?

(b) Now assume that the toy soldier is dropped (starting from rest) the instant the gun is

fired. If the gun is aimed at the toy soldier when it is fired, at what height above the

ground will the soldier be when it is hit by the bullet?

[pic]

(c) How long after firing does the bullet hit the soldier?

2B. The 20 kg mass is free to slide down an incline of 45º. The

pulley is massless and the 20 kg mass is connected to a 5 kg

mass as shown. Initially the string connecting the 20 kg mass

to the other is vertical. If the system is released from rest.

(a) Find the speed of the 5 kg mass when 20 kg mass has moved to

position “B” as shown in diagram if the incline was smooth. A

(b) If the plane was rough and (k = 0.21 what is the acceleration of the 20 kg mass

when it is at point A (assume (k = 0.21 for part c and d).

(c) Find the acceleration of the 20 kg mass at the position B, if the pulley was heavy and has

a moment of inertia about the center of mass (axis of rotation) of .25 kg m2.

(d) Find the tensions of the string at position B.

3B. A uniform wooden sign of mass 4.0 kg hangs beside a building wall. The sign is 2.00 m

high and 4.00 m wide. It is supported by a hinge at P, that is midway up one

edge, and by a light rope that is attached exactly three-quarters of the distance across the

upper edge. The rope makes an angle of 20.0 degrees with the horizontal.

(a) Complete the free body diagram by showing all the forces acting on the sign; show

where on the sign each of the forces acts.

(b) Write the conditions for static equilibrium of the sign and solve for the horizontal (H) and

vertical (V) components of the force at P, and solve for the rope tension (T)

Later, Prof. Fich of mass 90.0 kg climbs out of a window just below the sign, reaches up

and grabs onto the bottom portion of the sign and, while hanging from the sign, begins to move away from the building. If the rope can only support a maximum tension of 1500 N, how far can Prof. Fich move away from the building ?

4B. A uniform rod of mass m and length ( is pivoted at one end is free to rotate in a

vertical plane. There are 2 disks each of mass m and radius r attached to the rod

as shown in the diagram. A rubber bullet of mass mb is shot at the rod and it

sticks to the rod at a distance h from the pivot. If the speed of the bullet just before

it collides with the rod is 50 m/s, m = 2 kg, ( = 1m, r = 20 cm, mb = 0.5 kg

and h = 80 cm.

(a) Find the linear momentum of the bullet just before impact.

(b) Find the angular momentum of the system just before impact.

(c) Find the moment of inertia of the system after the bullet has collided.

(d) Find the angular velocity of the system after collision.

(e) Find the angle the system will rotate before coming to a momentary stop.

5B. Spheres A (mass 0.020 kg), B (mass 0.030 kg), and C (mass 0.050 kg), are each

approaching the origin as they slide on a frictionless air table as shown in the diagram.

The initial velocities of A and B are given in the figure. All three spheres arrive at the origin at the same time and stick together.

(i)

(a) What must the x- and y-components of the initial velocity of C be if all three objects are

to end up moving at 0.50 m/s in the +x-direction after the collision?

(b) If C has the velocity found in part (a), what is the change in the kinetic energy of the

system of three spheres as a result of the collision?

(ii) The triangle ABC is made from a uniform metal sheet. Like every triangle, the center of

mass of ABC is at a point on the line AD and is 2/3 AD when measured from apex A. If

a portion DEF is cut off from the triangle ABC where will the new centre of mass be?

Note: AG = 1 m

EF = 2m

BC = 6 m

The angles of ABC are 60º.

6B.

A cue hits the billiard ball at the position shown. The collision lasts 90 ms with an

average force of 420 N. If R = 7 cm, m = 300g, (k = 0.1.

(ignore friction during impact)

(a) Find the angular speed immediately after the collision.

(b) Find the linear speed of the centre of mass immediately after the collision.

(c) How much energy is lost to friction?

(d) How many revolutions does the ball go through before it begins to roll perfectly?

Blank Page

Kinematics [pic]

[pic]

[pic][pic]

[pic]

valid in both x and y directions

[pic]

[pic]

[pic]

Relative Velocity [pic]

Dynamics

Newton’s Second Law [pic]

Newton’s Third Law [pic]

Friction [pic]

[pic]

Circular Motion [pic]

[pic]

[pic]

Kinetic Energy & Work [pic]

[pic]

Work done by a spring [pic] or

[pic]

x = extension

Power P [pic]

[pic]

Potential Energy and Conservation of Energy

[pic]

[pic] x = extension

Conservation of mechanical energy

[pic]

Work by Nonconservative Forces

[pic]

Center of mass [pic]

[pic]

[pic]

Linear Momentum [pic]

[pic]

Conservation of Linear Momentum

[pic] (closed isolated system)

Variable Mass Systems [pic]

[pic]

Collisions [pic]

[pic]

Calculate [pic]

Calculate [pic] rel to cm

momentum rel to cm is zero

Rotation [pic]

[pic]

Constant angular acceleration [pic]

[pic]

[pic]

Circular motion [pic]

[pic]

Moment of inertia [pic]

Parallel axis theorem [pic]

TORQUE [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Rolling, Torque & Angular Momentum

Rolling bodies [pic]

[pic]

Torque as a vector [pic]

Angular Momentum [pic]

[pic]

[pic]

[pic] (rigid body, fixed axis)

Conservation of Angular Momentum

[pic]=constant for isolated system

[pic] (isolated system)

Static Equilibrium [pic]

[pic]

-----------------------

( =

h =

T =

v =

a =

a =

T1 =

T2 =

H =

V =

T =

p =

Li =

I =

(f =

( =

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