Question

A crate is at rest on an inclined plane. As the slope increases the crate remains at rest until the incline reaches an angle of 32.7° from the horizontal. At this angle the crate begins to slidedown the ramp.Apply Newton's 1st or 2nd law to each axis of the Free body diagram.

Answer 1

The free body diagram of the crate can be shown as,

According to free body diagram, the net force acting on the crate is,

`F=mg\sin \theta-f\ldots\ldots\text{ (1)}`

The frictional force acting on the crate can be given as,

`f=\mu N`

According to free body diagram, the normal force acting on the crate is,

`N=mg\cos \theta`

Therefore, the frictional force becomes,

`f=\mu mg\cos \theta`

According to Newton's second law, the net force acting on the crate is,

`F=ma`

Therefore, equation (1) becomes,

`\begin{gathered} ma=mg\sin \theta-\mu mg\cos \theta \\ a=g\sin \theta-\mu g\cos \theta \end{gathered}`

Thus, the net acceleration of the crate can be expressed using Newton's second law as,

`a=g\sin \theta-\mu\cos \theta`