Question

If a pushing force an angle $\alpha$ with horizontal is applied on a block of mass m placed on a horizontal table and angle of function is $\beta$, then what is the minimum magnitude of force required to move?

Answer 1

Here, we will use the basic concept of force. On an object with some mass and force applied on the object, the net force will be calculated by taking into account all the external forces acting on an object like- weight, force due to gravity. This will further help us in getting the required relation.
Formula used:
$\mu = \tan \beta$
$N = mg + F\cos \alpha$

Complete answer: As we know, there is always a normal force N acting on an object, which is perpendicular to the force applied on the object. Also, there is the presence of weight which is also taken into account.
Here, the minimum magnitude of force that is required to move the given block is expressed as
$\mu = \tan \beta$
$N = mg + F\cos \alpha$
\eqalign{ & F\sin \alpha = \mu N \cr & \Rightarrow F\sin \alpha = \mu [mg + Fcos\alpha ] \cr & \Rightarrow F[sin\alpha - \mu cos\alpha ] = \mu mg \cr & \Rightarrow F = \dfrac{{\mu mg}}{{\sin \alpha - \mu \cos \alpha }} \cr & \Rightarrow F = d\dfrac{{\tan \beta (mg)}}{{\sin \alpha - \tan \beta \cos \alpha }} \cr & \Rightarrow F = \dfrac{{\sin \beta (mg)}}{{\sin \alpha \cos \beta - \sin \beta \cos \alpha }} \cr & \therefore F = \dfrac{{\sin \beta (mg)}}{{\sin [\alpha - \beta ]}} \cr}
Therefore, here F represents the minimum amount of force required to move the block.

Additional information:
Force is simply any push or pull. The S.I unit of force is Newton represented by N. The acceleration is defined as the increase in the velocity of an object. The acceleration is measured in meters per Second Square.
There are three laws of motions given by Newton. These laws of motion relate an object's motion to the forces acting on it.
First law of motion states that an object continues to be in rest or in motion in a particular direction until and unless any external force is applied on it.
In the second law of motion, the force on an object is equal to its mass times its acceleration. This law also gives the relation of momentum and force.
In the third law of motion, every action has an equal and opposite reaction.

Note:
Force applied on an object changes its motion, speed, direction and also its shape. There is a very small difference between the force applied and the pressure applied on an object. Pressure is always given as the force per unit area.