Some Particular Forces — Question 5

Question 5

A normal force follows the surface normal

A 4 kg block slides on a fixed frictionless plane inclined $30^\circ$ above horizontal. Take $g=10$ m/s$^2$. There are no other forces.

1. Find the normal force.

2. Find acceleration along the plane.

3. Explain why N is neither mg nor a vertical force.

Question 5 - Solution

A normal force follows the surface normal

(a) Use zero perpendicular acceleration.

$$\sum F_\perp=N-mg\cos\theta=0\ \Rightarrow\ \boxed{N=40\cos30^\circ=20\sqrt3\,\mathrm N}$$ $N=mg\cos30^\circ=20\sqrt3\approx34.641$ N.

(b) Use the parallel gravitational component.

$$ma_\parallel=mg\sin\theta\ \Rightarrow\ a_\parallel=10\sin30^\circ=5\,\mathrm{m/s^2}$$ $ma_{\parallel}=mg\sin30^\circ$, so acceleration is 5 m/s$^2$ down the plane.

(c) Use contact geometry.

$$\vec N=N\hat n,\quad\vec W=mg\sin\theta\,\hat t-mg\cos\theta\,\hat n$$ The normal is perpendicular to the inclined surface. It cancels only gravity’s component into that surface; the remaining component accelerates the block downhill. Treating N as vertical would change the physical direction of the contact force.