A block of mass m = 5 kg starts from rest at the top of a frictionless, icy circ

Question

A block of mass m = 5 kg starts from rest at the top of a frictionless, icy circular quarter pipe of radius R = 2.0 m. When let go, it slides down to the bottom of the slope, and continues horizontally, until it encounters a rough patch of ice and gravel of length 1.0 m, with a coefficient of kinetic friction between the block and the patch equal to 0.30. It then continues on frictionless ice until it reaches a spring attached to a wall, of spring constant k = 100 N/m. As the block compresses the spring, it slowly comes to rest. How much does the block compress the spring?

Explanation / Answer

m = 5kg, R = 2 m , L = 1 m

uk= 0.3, k = 100 N/m

from conservation of energy

0.5mv^2 = mgR

work done by friction on the rough patch of length

W = -uk*mg*L

from work energy thorem

K = mgR - uk*mg*L

The spring compresses until all kinetic energy is transformed to potential energy in the spring:

0.5kx^2 = K

0.5kx^2= mgR - uk*mg*L

0.5*100*x^2 = 5*9.8*2 - 0.3*5*9.8*1

x = 1.3 m

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