Question

In the figure shown , angle of repose is ${45}^0$. Find the force of friction, net force and acceleration of the block when:
1) $\theta ={30}^0$
2)1) $\theta ={45}^0$
3)1) $\theta ={60}^0$

Answer 1

angle $g^{\prime }$ repose = the angle which the incline makes with horizontal when the block placed over it just starts moving

So in that case

Refer above image

$N=mg\cos \theta$

$fs=umg\cos \theta$

$mg\sin \theta =umg\cos \theta$

$\tan \theta =u$

So $\tan 54=u$

$\Rightarrow u=1$

Now at $\theta ={30}^0$ the block will not slide

at angle $\theta ={45}^0$ the block is just about to slide but will not move

and at angle $\theta ={60}^0$

$mg\sin 60-\left(1\right)\left(mg\right)\cos 60=ma$

$g\frac{\sqrt{3}}{2}-\frac{g}{2}=a$

$5(\sqrt{3}-1)=1$

$\Rightarrow a=3.66m/s^2$