A 1 kg block is pushed 10 m up a 30° incline by a 10 N force parallel to the plane with kinetic friction coefficient 0.1. What is the work done against gravity, the work done against friction, and the block’s acceleration up the plane?

What you need to calculate

Moving up an incline, the push force $F$ must overcome two forces along the plane: the downslope component of gravity $mg\sin\theta$ and kinetic friction $f_k = \mu N$. Once you know those, you can find the work done against each and then use Newton’s second law for acceleration.

Work done against gravity

The vertical height gained is $h = d\sin\theta$, so the increase in gravitational potential energy is $$ W_g = mgh = mg(d\sin\theta). $$ Substitute $m=1\ \text{kg}$, $g=9.8\ \text{m/s}^2$, $d=10\ \text{m}$, $\theta=30^\circ$: $$ W_g = (1)(9.8)(10)\sin 30^\circ = 98\cdot 0.5 = 49\ \text{J}. $$

Work done against friction

Because $F$ is parallel to the plane, it does not change the normal force. The normal force is $$ N = mg\cos\theta. $$ So kinetic friction is $$ f_k = \mu N = \mu mg\cos\theta. $$ Then the work done against friction over distance $d$ is $$ W_f = f_k d = \mu mg\cos\theta\, d. $$ Numerically: $$ W_f = 0.1(1)(9.8)\cos 30^\circ(10) = 0.98\cdot 0.866\cdot 10 \approx 8.49\ \text{J}. $$

Acceleration up the incline

Sum forces along the incline (up the plane is positive):

• Applied force: $+F$
• Gravity component down the plane: $-mg\sin\theta$
• Friction down the plane (opposes upward motion): $-\mu mg\cos\theta$

So the net force is $$ F_{\text{net}} = F - mg\sin\theta - \mu mg\cos\theta, $$ and $$ a = \frac{F_{\text{net}}}{m}. $$ Compute: $$ mg\sin 30^\circ = 9.8\cdot 0.5 = 4.9\ \text{N}, $$ $$ \mu mg\cos 30^\circ = 0.1\cdot 9.8\cdot 0.866 \approx 0.8487\ \text{N}, $$ $$ a = \frac{10 - 4.9 - 0.8487}{1} \approx 4.25\ \text{m/s}^2. $$

Quick check (does it make sense?)

Since $F=10\ \text{N}$ is larger than the total resisting force $4.9 + 0.85 \approx 5.75\ \text{N}$, the net force is positive, so the block accelerates upward, which matches the result.