Work Done on a System by an External Force — Question 3

Question 3

Uniform ascent with internal frictional heating

Assume no heat transfer across the stated system boundary during the process unless otherwise specified. A 3 kg crate is pulled at constant speed up a fixed incline for 4 m, rising vertically 2 m. Friction is 5 N. Choose crate, ramp and Earth as the system; take g=10 m/s$^2$.

1. Find changes in kinetic, gravitational and thermal energies.

2. Find external pulling work.

3. Find the constant pulling force if it acts parallel to the incline.

Question 3 - Solution

Uniform ascent with internal frictional heating

(a) Identify all changing stores.

Constant speed eliminates the kinetic change, but not the other stores: $$\boxed{\Delta K=0},\qquad\boxed{\Delta U_g=3(10)(2)=60\ \mathrm J},$$ $$\boxed{\Delta E_{\mathrm{th}}=5(4)=20\ \mathrm J}.$$

(b) Sum the energy requirements.

The external input equals the total storage increase: $$\boxed{W_{\mathrm{pull}}=\Delta K+\Delta U_g+\Delta E_{\mathrm{th}}=0+60+20=80\ \mathrm J}.$$

(c) Relate the external work to its own displacement.

The force acts along its 4 m displacement: $$\boxed{F_{\mathrm{pull}}=W_{\mathrm{pull}}/s=80/4=20\ \mathrm N}.$$ Force balance confirms $mg\sin\theta+f=30(2/4)+5=20$ N.