Question

A 3 kg mass starts from rest at point A, which is at the top of an h= 0.50 m high, frictionless curved ramp. It slides down the ramp, encounters a surface which has a coefficient of friction of μ=0.1 for a length of 1.0 meters. Once the mass hits point C, the surface becomes frictionless. Finally, the mass encounters a spring (k=100 N/m ), compressing the spring, until the mass comes momentarily to a stop. Use conservation of energy to answer each question. (a) What is the magnitude of the velocity of the mass at point B? (b) What is the magnitude of the velocity of the mass at point C? (c) How much does the spring compress when the mass comes momentarily to a stop? (d) After stopping for a moment, the spring expands sending the mass back towards point C. What is the magnitude of the velocity at Point C?

Answer 1

Answer:

a) 3.13 m/s

b) 2.8 m/s

c) 0.485 m

d) 2.8 m/s

Explanation:

a) PE @ A = KE @ B

mgh = ½ mv²

v = √(2gh)

v = √(2 × 9.8 m/s² × 0.50 m)

v = 3.13 m/s

b) PE @ A = KE @ C + Work done by friction

mgh = ½ mv² + Fd

mgh = ½ mv² + mgμd

gh = ½ v² + gμd

g (h − μd) = ½ v²

v = √(2g (h − μd))

v = √(2 × 9.8 m/s² (0.50 m − 0.1 × 1 m))

v = 2.8 m/s

c) KE @ C = EE

½ mv²  = ½ kx²

mv²  = kx²

x = v √(m / k)

x = (2.8 m/s) √(3 kg / 100 N/m)

x = 0.485 m

d) Since there's no friction between point C and the spring, the speed is the same as before.

v = 2.8 m/s