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Name _____________________ NetID_____________ Starting Time ______________
You have one hour to work on following practice exam. If you cannot solve any problems, please move
on to the next problem. Show your work on the problem sheets and note your final answers again on this
sheet for quick checking.
1. Work done on the box by friction is _____________ J
2. The maximum extension of the spring is _____________ m
3. The speed of the mass at the equilibrium is _____________ m/s
4. The speed of the comet at 2.25x10^11 m from the sun is ______________ m/s
5. a) h 2 = h 1 b) h 2 > h 1 c) h 2 < h 1
6. The maximum radius (in term of given variables) is _____________
7. a) Block A b) Block B c) Both blocks end up with the same momentum.
8. a) Block A b) Block B c) Both blocks end up with the same momentum
9. The interation time is _____________ s
10. The speed of the car after the shell is fired is _____________ m/s
11. The horizontal position of the center of mass is at x = _____________ m
12. The trailer moves _____________ m
13. Velocity of the center of mass is _____________ m/s
14. The common velocity (in term of given variables) when the spring is maximally compressed is
_____________
15. The maximum compression of the spring is _____________ m
16. The kinetic energy ratio, KE 1 / KE 2 , (in term of given variables) is _____________
Solution 1st^ set
1. Wf = -5.17 J
2. xmax = 2.3 m
3. vequilibrium = 0.55 m/s
4. vf = 23700 m/s Work and Conservation of Energy
5. b) h 2 > h 1
〣ㄘ ⡳〴
7. c) Both blocks end up with the same momentum
8. b) Block B ends up with the biggest kinetic energy
9. tf = 0.0129 s Problems on Impulse
10. vx, car+cannon = 2.67 m/s
11. xcm = 0.58 m
12. ∆x = 4/3 m Center of mass & cons of momentum
〔ㄗ〣ㄗ 〔ㄗ⡸〔ㄘ
15. dmax = 0.266 m
16. 〒〆 〒〆ㄗㄘ = 〔 〔ㄘㄗ Combination
- Using Work-Kinetic Energy Theorem (∆K = WNet), where net work consists of work done by friction (Wf) and work done by the rope tension (WT), we have: 1 2 ᡥᡴ〳
⡰ −^1
⡰ −
⡰ = 䙦5 ᡀ䙧 ∙ 䙦2 ᡥ䙧 + ᡉ〳
ᡉ〳 = −5.17 ᠶ
- In this problem, the energy is conserved. When the spring is maximally extended, the kinetic energy is zero and all the change in gravitational potential energy is stored in the spring.
ᠷᠱ〶 + ᡂᠱ〴,〶 + ᡂᠱうぃぅ〶ぁ〴,〶 = ᠷᠱ〳 + ᡂᠱ〴,〳 + ᡂᠱうぃぅ〶ぁ〴,〳
ᠷᠱ〶 = ᠷᠱ〳 = 0
∆ᡂᠱ〴 = −ᡥᡙ䙦sin 30 ∙ ᡶ䙧
Thus,
∆ᡂᠱ〴 + ∆ᡂᠱうぃぅ〶ぁ〴 = 0
1 2 ᡣᡶ⡰^ − ᡥᡙ䙦sin 30 ∙ ᡶ䙧 = 0
2ᡥᡙ sin 30 ᡣ
2 ∙ 䙦0.7 ᡣᡙ䙧 ∙ 䙦9.8 ᡱᡥ⡰䙧 ∙ sin 30 䙦3 ᡀ/ᡥ䙧
3. A massless spring of spring constant k = 30 N/m hangs vertically in the earth's gravitational field. A 1 kg
mass is attached to the spring. Suppose now that the mass is pulled down from the equilibrium position a
distance of 0.1 m and released. What is the speed of the mass when it returns to the equilibrium
position?
4. A comet of mass 10^9 kg is observed at a distance from the sun of 8 × 10^11 m (mass of sun = 2 × 10^30 kg)
at a speed of 17000 m/s. Assuming no forces on it other than the sun's gravity, how fast will it be going
when it is a distance of 2.25 × 10^11 m from the sun? (The universal gravitational constant is G = 6.67 ×
10 -11^ Nm^2 /kg^2 )
5. Two identical blocks initially have the same velocity V at the bottom of two ramps. The first ramp
inclined at a shallower angle ( θ 1 ) with respect to the horizontal than the second ramp ( θ 2 ). The maximum
heights reached by the blocks are h 1 and h 2 respectively. Now assume instead that in both cases there is
the same (non-zero) coefficient of kinetic friction between the blocks and the ramps. Which statement is
correct concerning the maximum heights reached by the blocks?
a) h 2 = h 1 b) h 2 > h 1 c) h 2 < h 1
6. A cart on a track enters the bottom of a frictionless loop-the-loop at with a velocity V. What is the
maximum radius, R, the loop can have such that the cart does not fall off at the top?
g
V
R
- If the incline is frictionless, then both masses can move up the incline to the same height. However, since it is not frictionless and the friction on the incline depends on the angle,ᡘ = †ᡥᡙ cos ‖, both masses won’t move up to the same height.
From the problem situation, ᠷᠱ〶 = ᡘ ∙ ᡶ + ᡥᡙℎ = †ᡥᡙ cos ‖
sin ‖ + ᡥᡙℎ
䙦†ᡥᡙ cot ‖ + ᡥᡙ䙧
ᡅᡡᡦᡕᡗ 0 < ‖⡩ < ‖⡰ < 90°, ᡱᡧ ℎ⡩ < ℎ⡰
- The energy is conserved in this problem. The condition that the cart does not fall off at the top is that the required centripetal force is equal to or greater than mg.
Use conservation of energy to calculate velocity at the top of the loop, 1 2 ᡥᡈ
⡰ =^1
ᡴぇあぃ = 㒓ᡈ⡰^ − 2ᡙ䙦2ᡄ䙧
To calculate the maximum radius,
ᡥᡴぇあぃ⡰ ᡄ
ᡥ㐵ᡈ⡰^ − 2ᡙ䙦2ᡄ䙧㐹
7.&8. In this problem two blocks were pushed by identical forces for identical periods of time. So, both blocks received the same impulse.
ᠵᡥᡨᡳᡤᡱᡗ = ᠲ㍷〨ぉ〲ぅ〨〴〲∆ᡲ = ∆ᡂ䙒㍷
Since both blocks are initially at rest, then they receive the same impulse (change of momentum), they end up with the same momentum.
For kinetic energy, since both blocks have the same momentum, the smaller block must have a bigger velocity. These two conditions imply that smaller block must also have a bigger kinetic energy.
ᡥ。 = 2ᡥ〃
ᡥ。ᡈ。 = ᡥ〃ᡈ〃
ᡈ。 =
So block B has bigger kinetic energy than block A.
- In the inelastic collision case, the egg stick to the floor, i.e., final velocity and final momentum is zero. The change of momentum is equal to the impulse.
- In this problem the horizontal momentum is conserved.
0 = 䙦20 ᡣᡙ䙧 䙲−300 cos 27°
11. The centers of three spheres having masses 1 kg, 2 kg and 3
kg are placed at the corners of an equilateral triangle whose
sides are each 1 meter long, as shown below. What is
the horizontal position of the center of mass?
12. A 1000 kg horse trailer with frictionless wheels is sitting in a level parking lot. The trailer is 4 m long,
and its center of mass is at its center. Its passenger, a 500 kg horse, breaks free from its stall at one end
of the trailer and walks to the other end. How far does the trailer move relative to the ground? Treat the
horse as a point particle. The mass of the trailer above does not include the 500 kg horse.
13. A 12 kg block moves in the x -direction at 28
m/s, and a 24 kg block moves in the y -
direction at 8.0 m/s. Find the velocity of their
center of mass.
14. A block of mass M 1 slides on a horizontal, frictionless surface with a velocity of V 1. It collides with an
ideal, massless spring which is attached to a block of mass M 2 which is initially at rest. The spring has a
spring constant of k = 20 N/m. At the instant the spring is maximally compressed, both masses will be
traveling with a common velocity, V , of?
- Velocity of the center of mass can be calculated as follows.
ᡴ㍷〰 =
- Since there is no external force, the momentum is conserved at all time during the collision.
㔳 ᡂ䙒㍷〶 = 㔳 ᡂ䙒㍷⤣⤑⤴ 〰あぃぅ〲うう〲〱
Given that both masses travel with the same velocity when the spring is maximally compressed,
ᠹ⡩ᡈ⡩ = 䙦ᠹ⡩ + ᠹ⡰䙧ᡈ
- This is a two-step problem, inelastic collision first then energy transfer. During the collision, the momentum is conserved, but the energy is not conserved. During the energy transfer into the spring, the energy is conserved. First process, inelastic collision (12) 㔳 ᡂ⡩ = 㔳 ᡂ⡰ ᡥᡴ = 䙦ᡥ + ᠹ䙧ᡈ⡰
ᡈ⡰ =
Second process, energy transfer (23) 㔳 ᠱ⡰ = 㔳 ᠱ⡱ 1 2
䙦ᡥ + ᠹ䙧ᡈ⡰⡰^ =
⡰ ᡖ〨け = 0.266 ᡥ
- Momentum is conserved in this problem.
㔳 ᡂ〶 = 㔳 ᡂ〳 0 = ᠹ⡩ᡴ⡩ + ᠹ⡰ᡴ⡰
㘨
ᡴ⡰^ 㘨 =
Thus,
⡰ 1 2 ᠹ⡰ᡴ⡰
⡰
⡩
⡰
ᠹ⡰ᡴ⡰⡰^
