A 4.5 kg box slides down a 5.1-m -high frictionless hill, starting from rest, ac

Question

A 4.5 kg box slides down a 5.1-m -high frictionless hill, starting from rest, across a 2.3-m -wide horizontal surface, then hits a horizontal spring with spring constant 460 N/m. The other end of the spring is anchored against a wall. The ground under the spring is frictionless. but the 2.3-m-long horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.23. 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. What is the speed of the box just before hitting the spring? Express your answer to two significant figures and include the appropriate units. How far is the spring compressed? Express your answer to two significant figures and include the appropriate units. Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?

Explanation / Answer

A)from the conservation of energy

1/2 m v^2 = m g h

v = sqrt (2 g h)

v = sqrt (2 x 9.8 x 5.1) = 10 m/s

Hence, v = 10 m/s

B)Again from conservation of energy:

KEi - Wfric = KEf

1/2 m v2^2 = 1/2 m v1^2 - uk mg x

0.5 v2^2 = 0.5 x 10^2 - 0.23 x 9.8 x 2.3

v2 = 9.5 m/s

C)KE before the spring is compressed will be given by:( v = 10)

KE(before spring) = KE(at bottom) - Work done by friction

W(f) = uk M g d = 0.23 x 4.5 x 9.8 x 2.3 = 23.33 J

KE(bottom) = 0.5 m v2 = 0.5 x 4.5 x (10)^2 = 225 J

KE (before spring) = 225 - 23.33 = 201.67 J

This much amount of energy is stored at maximum spring compression. So

KE(before soring) = PE of the spring = 1/2 k x^2

1/2 k x^2 = 201.67

x = sqrt (201.67 x 2 / 460) = 0.94 m

x = 0.94 m

(4) each time the box is crosses the rough surface it is loosing its stored PE by W(f)

intital energy of box = PE (at top ) = 4.5x 9.8 x 5.1 = 225 J

The box will come to rest after it will loose all its stored energy. So

No of trips = 225/23.33 = 9.64

Hence it will complete 10 complete trips. (or 9)

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