Question

A 4.5 kg box slides down a 4.8-m -high frictionless hill, starting from rest, across a 2.0-m -wide horizontal surface, then hits a horizontal spring with spring constant 520 N/m . The other end of the spring is anchored against a wall. The ground under the spring is frictionless, but the 2.0-m-long horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.24.

Part A

What is the speed of the box just before reaching the rough surface?

Express your answer to two significant figures and include the appropriate units.

Part B

What is the speed of the box just before hitting the spring?

Express your answer to two significant figures and include the appropriate units.

Part C

How far is the spring compressed?

Express your answer to two significant figures and include the appropriate units.

Part D

Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?

Answer 1

A)PE at the top will be:

PE = m g h = 4.5 x 9.8 x 4.8 = 211.68 J

from energy conservation:

KE(bottom) = PE(top)

1/2 m v^2 = 211.68

v = sqrt (2 x 211.68/4.5) = 9.7 m/s

Hence, v = 9.7 m/s

b)Work done by the froctional force is:

Wf = Ff x d

Wf = (mu m g) d

Wf = 0.24 x 4.5 x 9.8 x 2 = 21.17 J

Ke just before hitting is:

KE = Ke(bottom) - Wf

KE = PE(top) - Wf = 211.68 - 21.17 = 190.51

1/2 m v'^2 = 192.51 J

v' = sqrt (2 x 190.51/4.5) = 9.2 m/s

Hence, v' = 9.2 m/s

C)again from conservation of energy

1/2 m v^2 = 1/2 k x^2

x = v sqrt (m/k)

x = 9.2 sqrt (4.5/520) = 0.86 m

Hence, x = 0.86 m

D)Each time box crosses the frictional surface it looses 21.17 J of energy so number of trips would be:

N = PE(top)/Wf = 211.68/21.17 = 9.99 = 10

Hence, N = 10