Physics 110

Spring 2009

Exam #2

May 13, 2009

Name______________

In keeping with the Union College policy on academic honesty, it is assumed that you will neither accept nor provide unauthorized assistance in the completion of this work.

Part Multiple Choice / 10

Problem #1 / 27 Problem #2 / 36 Problem #3 / 27

Total / 100

k

Part I: Free Response Problems Please show all work in order to receive partial credit. If your solutions are illegible

no credit will be given. Please use the back of the page if necessary, but number the problem you are working on. Each subpart of a problem is worth 9 points.

1. A cannon is rigidly connected to a carriage, which can move along a frictionless

horizontal surface but is connected to a wall by a large spring with stiffness k = 2.0x104 N/m. The cannon can fire 200-kg projectiles at a velocity of 45-m/s directed at 45o above the horizontal.

a. If the mass of the carriage and cannon is 5000-kg, what is the recoil speed of the cannon?

€

pix = pfx → 0 = −mC +CVrecoil + mballv cosθ

∴vrecoil =mballv cosθmC +C

=200kg× 45 m

s cos455000kg

=1.27 ms

b. Using energy methods, what is the maximum extension of the spring?

€

ΔKE+ ΔUs = 12mC+Cvf

2 − 12mC+Cvi

2( ) + 12 kxf

2 − 12 kxi

2( ) = − 12mC+Cvi

2 + 12 kxf

2 = 0

xf =mC+Cvi

2

k=

5000kg× 1.27 ms( )2

2.0×104 Nm

= 0.64m

c. What is the maximum force exerted on the carriage by the spring?

€

Fmax = kxmax = 2.0×104 Nm × 0.64m = 12,700N to the right.

2. A block of mass m slides down a curved frictionless track and then up an incline. The coefficient of kinetic friction between the block and the incline is µk.

a. Using energy methods, if the block is released from rest, what is its speed as it

traverses the horizontal portion of the track?

€

ΔUg + ΔKE = mgyf − mgyi( ) + 12mvf

2 − 12mvi

2( ) = −mgh+ 12mvhoriz

2 = 0

∴ vhoriz = 2gh

b. What is the maximum vertical height ymax reached by the block?

€

ΔUg + ΔKE = −Ffrd → mgymax − mgyi( ) + 12mvf

2 − 12mvi

2( ) = − µ kmgcosθ( ) ×ymaxsinθ

∴ ymax =h

1+ µ k cotθ( )

c. Suppose that as the block is traveling to ymax someone places a second block of equal mass at rest in the middle of the horizontal portion of the track. When the block on the incline reaches ymax it momentarily comes to rest before sliding back down the ramp at which point it makes an inelastic collision with the second block. What is the speed of the two blocks in terms of g, ymax, µk, and θ?

€

pix = pfx → mvhoriz = m + m( )V →V =vhoriz2

=2gymax 1−µk cotθ( )

2where

€

ΔUg + ΔKE = −FFRd→ mgy f −mgymax( ) + 12mvhoriz

2 − 12mvi

2( ) = −µkmgcosθ ×ymaxsinθ

vhoriz = 2gymax 1−µk cotθ( )

d. To what height to up the curved portion of the ramp do the two blocks reach, in terms of ymax, µk, and θ?

€

ΔUg + ΔKE = 2mgyup − 2mgyi( ) + 12 2mv f

2 − 12 2mV

2( ) = 0

yup =V 2

2g=ymax 1−µk cotθ( )

4

3. A model airplane of mass 0.75 kg flies in a horizontal circle at the end of a 60 m control wire with a speed v. The control wire makes an angle of 20o below the horizontal and the wings of the airplane produce a lifting force of 14.2 N that is perpendicular to the wings.

a. What is the tension force in the control wire?

€

Fx : − FL sinθ∑ − FT cosθ = −mv 2

RFy : FL cosθ∑ − FT sinθ −mg = 0

→ FT =FL cosθ + mg

sinθ=14.2N cos20 − 0.75kg× 9.8 m

s2

sin20=17.5N

b. What is the speed of the airplane?

c. Suppose that you release the control wire so that it is 20 m longer and that the

tension in the wire and the angle the control wire makes with the horizontal remain constant. What is the new speed of the airplane? (Hint: The lifting force does not necessarily remain constant.)

60m

20o

Part II: Multiple-Choice Circle the best answer to each question. Any other marks will not be given credit. Each multiple-choice question is worth 2 points for a total of 10 points.

1. A dart is loaded into a spring-loaded toy dart gun by pushing the spring in by a

distance d. For the next loading, the spring is compressed a distance 2d. How much work is required to load the second dart compared to that required to load the first?

a. It takes four times as much work. b. It takes two times as much work. c. It takes the same amount of work. d. It takes half as much work.

Questions 2 & 3 refer to the conical pendulum shown on the right in which a bob of mass m is spun at an angle θ and the bob traces out a circle of radius R in the horizontal plane. 2. From Newton’s 2nd law, the tension force in the string is

a. b.

c. d.

3. The speed of the bob in the horizontal plane is

a. b. c. d. 4. Suppose that a bowling ball and a baseball are thrown off of a high building with the

same magnitude of the velocity. Let the bowling ball be thrown horizontally while the baseball is thrown upward at an angle θ with respect to the horizontal. Ignoring air resistance, the balls a. have the same magnitude of the velocity at the bottom. b. vbaseball > vbowling ball. c. vbowling ball > vbaseball. d. cannot tell from the information given.

5. A Ping-Pong ball is thrown at a stationary bowling ball hanging from a wire. The Ping-Pong ball makes a one-dimensional elastic collision and bounces back along the same line. After the collision, the Ping-Pong ball has, compared with the bowling ball,

a. a larger magnitude of momentum and more kinetic energy. b. a smaller magnitude of momentum and more kinetic energy. c. a larger magnitude of momentum and less kinetic energy. d. a smaller magnitude of momentum and less kinetic energy.

Useful formulas:

Motion in the r = x, y or z-directions Uniform Circular Motion Geometry /Algebra

Vectors Useful Constants

Linear Momentum/Forces Work/Energy Heat

Rotational Motion Fluids Simple Harmonic Motion/Waves Sound