A 18 kg box slides from rest down a ramp inclined at 25° to the horizontal onto a spring with a spring constant 740 N/m
as shown in the diagram. The spring is compressed 0.32 m before the box stops. Determine how far along the ramp the
box will slide before it stops. Assume friction is negligible.
To determine how far along the ramp the box will slide before it stops, we need to use the principle of conservation of mechanical energy.
First, let's calculate the initial potential energy of the box at the top of the ramp and its final potential energy when it stops. We can use the formula: potential energy (PE) = mass (m) * gravity (g) * height (h).
Given:
Mass of the box (m) = 18 kg
Gravity (g) = 9.8 m/s^2
Height of the ramp (h) = unknown
At the top of the ramp, the box has only potential energy since it's at rest. So, the initial potential energy is given by:
PE_initial = m * g * h
Now, let's calculate the final potential energy. When the box stops, it compresses the spring by 0.32 m. The spring will store potential energy, which can be expressed as:
PE_final = 0.5 * k * x^2
Given:
Spring constant (k) = 740 N/m
Compression of the spring (x) = 0.32 m
Now, we can equate the initial potential energy to the final potential energy since mechanical energy is conserved:
m * g * h = 0.5 * k * x^2
Let's solve for h.
h = (0.5 * k * x^2) / (m * g)
Plugging in the values we have:
h = (0.5 * 740 N/m * (0.32 m)^2) / (18 kg * 9.8 m/s^2)
Using this equation, we can now calculate the distance along the ramp that the box will slide before stopping.