A 4 kg block is pulled at constant speed up a 30° incline (μ = 0.3). Find the pulling force along the plane — Physics problems | PhysLabo
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Pulling force for uniform motion up an incline

Medium 10 grade Mechanics / Newton's Laws
Problem statement

A block of mass $m = 4$ kg is pulled at constant speed up an incline of angle $\alpha = 30°$ by a force $F$ directed along the plane (see the figure). The coefficient of friction between the block and the plane is $\mu = 0.3$. Find the force $F$. Take $g = 10$ m/s².

A block on an incline of angle α; force F acts up along the plane; mg, N and friction are shown
Given
  • $m = 4$ kg
  • $\alpha = 30°$
  • $\mu = 0.3$
  • $g = 10$ m/s²
  • $a = 0$
  • $F = ?$
Hints
Hint 1 / 2
Uniform motion means $a = 0$, so the net force along the plane is zero. Friction now points down the slope — against the motion.
Hint 2 / 2
First find $N = mg\cos\alpha$ from the balance of forces perpendicular to the plane, then $F_{fr} = \mu N$.
Solution

The motion is uniform, $a = 0$, so the net force is zero. Take the $Ox$ axis along the plane (up the slope) and the $Oy$ axis perpendicular to the plane. Friction opposes the motion, so it points down the slope.

Resolving gravity along the plane and perpendicular to it
Resolving gravity along the plane and perpendicular to it

1. Perpendicular to the plane:

$$N - mg\cos\alpha = 0 \;\Rightarrow\; N = mg\cos\alpha = 40 \cdot 0.866 \approx 34.6 \text{ N}$$

2. Friction:

$$F_{fr} = \mu N = 0.3 \cdot 34.6 \approx 10.4 \text{ N}$$

3. Along the plane:

$$F - mg\sin\alpha - F_{fr} = 0$$$$F = mg(\sin\alpha + \mu\cos\alpha) = 40 \cdot (0.5 + 0.3 \cdot 0.866) \approx 30.4 \text{ N}$$
Answer

$F \approx 30.4$ N

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