Question

Find out the acceleration of a body of mass 'm' sliding up with an acceleration a along an inclined plane with angle of repose phi.Its urgent and I need step by step explaination.

Answer 1

Given:

Body of mass 'm' sliding up with an acceleration a along an inclined plane with angle of repose phi.

To find:

The value of acceleration of body ?

Calculation:

Angle of repose (or angle of sliding) is the minimum angle of inclination of the incline plane with the horizontal such that the body starts sliding downwards .

In this case , the angle of repose simply represents the the angle made by the incline plane with the horizontal.

Since the block is accelerating upwards, friction will act downwards along the plane.

Let the applied force be F:

According to FBD of the block :

$\therefore \: F - mg \sin( \phi) -f = ma$

$\implies \: F - mg \sin( \phi) - \mu mg \cos( \phi) = ma$

$\implies\: F - mg \bigg \{\sin( \phi) + \mu \cos( \phi) \bigg \} = ma$

$\implies\: a = \dfrac{F - mg \bigg \{\sin( \phi) + \mu \cos( \phi) \bigg \}}{m}$

$\implies\: a = \dfrac{F}{m} - g \bigg \{\sin( \phi) + \mu \cos( \phi) \bigg \}$

So, value of acceleration is:

$\boxed{ \bold{\: a = \dfrac{F}{m} - g \bigg \{\sin( \phi) + \mu \cos( \phi) \bigg \}}}$

Answer 2

Question:-

Find out the acceleration of a body of mass 'm' sliding up with an acceleration a along an inclined plane with angle of repose phi.

Answer:-

Angle of repose is the maximum angle of inclination between an incline and horizontal till which, an object placed on it stays at rest.

The angle of repose can range from 0° to 90°.

But since here an external force is applied upwards, the block instead of staying still, moves upward.

Refer to the attachment for F.B.D.

Here, $\sf F$ is the applied upward force, while $\sf f$ is the frictional force acting opposite to the direction of relative motion (downward direction).

$\sf F -f - mgsin \phi = ma$

$\longrightarrow \sf F - \mu mgcos \theta \: - mgsin \theta = ma$

$\longrightarrow \sf F - mg( \mu cos\phi \: + sin \phi) = ma$

$\longrightarrow \sf a = \dfrac{F - mg(\mu cos \phi + sin\phi)}{m}$

$\longrightarrow \underline{\underline{\sf{\green{a = \dfrac{F}{m} - g( \mu \: cos\phi + sin \phi)}}}}$