Answer to Question #156957 in Classical Mechanics for Emre

Question

A box of mass m=15 kg at the packaging section of a factory comes to the top of a ramp ($\theta =37^{\circ }$) with speed vo and slides down where it is picked up for shipment. In order to avoid damage to the box a spring is used with force constant k=100N/m and the maximum force Fmax=100N. The box slides a distance of l=4 m down the incline before it hits the spring as shown. The coefficient of kinetic friction between the box and entire ramp is 0.75.

a) Find the work done by the gravity , normal force and friction force on the box until it hits the spring.

b) Find the maximum speed of the box at the top of the ramp if the box is to be picked up when the spring is in maximum compression.

Answer 1

a)

$$W_g=mgl\sin{37}=(15)(9.8)(4)\sin{37}=353.9\ J \\W_N=0\ J \\W_f=-\mu mgl\cos{37}\\=-0.75(15)(9.8)(4)\cos{37}=-352.2\ J$$

b)

$$x=\frac{F}{k}=\frac{100}{100}=1\ m$$

$$0.5mv^2+W_g+W_f=0.5kx^2\\0.5(15)v^2+353.9-352.2=0.5(100)(1)^2\\ v=2.5\frac{m}{s}$$