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A block of mass 1 kg lying on the floor is subjected to a horizontal force given by $ F = 2sin\omega t $ Newtons. The coefficient of friction between the block and the floor is $ 0.25 $ . The acceleration of the block will be:
A) positive and uniform
B) positive and non-uniform
C) zero
D) depending on the value of $ \omega $

Last updated date: 20th Jun 2024
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Hint: In this solution, we will first check for the maximum friction force that will act on the object and the maximum force that can be exerted on the object. The external force has to be greater than the friction force if the object has to be accelerated.

Formula used: In this solution, we will use the following formula
 $ {F_f} = \mu mg $ where $ \mu $ is the coefficient of friction, $ m $ is the mass of the block, and $ g $ is the gravitational acceleration.

Complete step by step answer
We’ve been given that the force acting on the block is $ F = 2sin\omega t $ . Let us start by finding the maximum value of $ F $ . The value of $ F $ changes with time and the maximum value corresponds to $ \sin \omega t = 1 $ which is
 $ F = 2 $
Now, the friction force acting on the object will be
 $ {F_f} = 0.25 \times 1 \times 9.8 $
 $ \Rightarrow {F_f} = 2.45\,N $
So we can see that friction force is greater than the external force acting on the object. Hence for no value of the external force will the body be accelerated.
Hence the correct choice is option (C).

For the object to accelerate the external force must exceed $ 2.45\,N $ in which case it can overcome the friction force and be accelerated. The tricky part in this question is realizing that the maximum value of the external force will only depend on the maximum value of the sine function. This is because regardless of the angular frequency and the time of the external force, the maximum value it can ever attain is 1. All the other values of the forces at different times will be lower than $ 2\,N $ and hence the block can never be accelerated.