Question

A 195 g block is pressed against a spring of force constant 1.60 kN/m until the block compresses the spring 10.0 cm. The spring rests at the bottom of a ramp inclined at 60.0° to the horizontal. Using energy considerations, determine how far up the incline in m) the block moves from its initial position before it stops under the following conditions. 

(a) if the ramp exerts no friction force on the block 

(b) if the coefficient of kinetic friction is 0.360 m 

(c) What If? If the ramp is 4.00 m long, what is the maximum coefficient of friction that would allow the block to reach the end of the ramp?

Answer 1

here,

mass of block , m = 195 g = 0.195 kg

the force constant , K = 1.6 KN /m

K = 1600 N/m

compression in the spring , x = 10 cm = 0.1 m

theta = 60 degree

a)

let the distance traveled up the incline be s

using Work-energy theorm

0.5 * k * x^2 = m * g * s * sin(theta)

0.5 * 1600 * 0.1^2 = 0.195 * 9.81 * s * sin(60)

solving for s

s = 4.83 m

the distance traveled is 4.83 m

b)

the coefficient of kinetic friction , uk = 0.36

let the distance traveled up the incline be s

using Work-energy theorm

- uk * m * g * cos(theta) * s = m * g * s * sin(theta) - 0.5 * k * x^2

- 0.36 * 0.195 * 9.81 * cos(60) * s = 0.195 * 9.81 * s * sin(60) - 0.5 * 1600 * 0.1^2

solving for s

s = 4.0 m

the distance traveled is 4.0 m

c)

when the distance traveled up the incline is 4 m

the maximum coefficient of friction is 0.36