Question

A box of mass m=40 kg is released from rest at the top of a frictionless ramp. The ramp makes an angle θ=45∘ to the horizontal. After sliding from the origin a distance d=2.00 m down the ramp, the box strikes a spring of force constant k=500 N/m, as shown at right. The box compresses the spring, momentarily stops, and then begins to slide back up the ramp.

What is the maximum speed of the box?

Answer 1

By using the conservation of energy principle, the box's gravitational potential energy at the top of the ramp converts into elastic potential energy when it compresses the spring, which then becomes kinetic energy as the box moves back up the ramp.

Step-by-step explanation:

Calculating the Maximum Speed of the Box

To solve the mathematical problem completely, we use energy conservation. Initially, the box has gravitational potential energy (GPE) since it is at rest at a height d*sin(θ) down the incline. As the box slides down, the GPE converts into kinetic energy (KE). Upon hitting the spring, all the box's KE changes into the elastic potential energy (EPE) of the spring at the moment when the box momentarily stops. When the box starts to move back up the ramp, this EPE changes back into KE.

Step 1: Calculate the box's initial GPE.

GPE = m*g*h = m*g*d*sin(θ) = 40 kg * 9.8 m/s2 * 2.00 m * sin(45°)

Step 2: Determine the maximum compression of the spring using the GPE.

EPE = GPE ⇒ ½*k*x2 = m*g*d*sin(θ)

Step 3: Calculate the maximum compression, x, of the spring.

x = √(2*m*g*d*sin(θ)/k)

Step 4: At the point of maximum compression, the box's speed is zero, but as it loses contact with the spring, all EPE has been converted back into KE.

KE = ½*m*v2 = EPE = ½*k*x2

Step 5: Calculate the maximum speed of the box as it starts to move back up the ramp.

v = √(2*EPE/m) = √(k*x2/m)

We do not need to account for the distance the box moves to find the maximum speed, as maximum speed occurs right before it makes contact with the spring again.