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Question

A body takes time t to reach the bottom of an inclined plane of angle θ will the horizontal. if the plane made rough, time takes now is 2t. The coefficient of friction of the rough surface is


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Solution

Step 1: Given data

  1. The angle of the inclined plan is θ.
  2. The initial velocity of the body is zero.
  3. Time is taken by the body to reach from top to bottom on a smooth inclined plane is t sec.
  4. Time is taken by the body to reach from top to bottom on a rough inclined plane is 2t sec.

Step 2: Exerted force on a body placed on an inclined plane

  1. For a block on an inclined plane, there are basically three types of force exerted, frictional force, normal force, and gravitational force.
  2. The frictional force is a force, that is directed opposite to the body's motion and that prevents the movement of the body at the point of contact.
  3. The frictional force is defined by the form, F=μN, where μis the coefficient of friction and n is the normal force.

Step 3: Diagram

A body takes time t to reach the bottom of an inclined plane of angle theta  will the horizontal. if the plane made rough, time takes now is 2t . The  coefficient

Step 4: Finding the coefficient of friction for the smooth inclined plane

the distance (hypotenuse) between the top point and bottom point is, ssmooth=12gsinθ×t2 ……………..(1)

(applying the formulae of kinematics)where g is the acceleration due to gravity and t is the taken time.

Step 5: Finding the coefficient of friction for the Rough inclined plane

the normal force (N) on the body is,

N=mgcosθ ……………(2)

Now, the frictional force of the body on the rough plane is,

F=υN=μmgcosθorF=μmgcosθ...................(3)

And the acceleration of the body is,

a=gsinθ-μgcosθora=g(sinθ-μcosθ)................(4)

Again,

srough=12sinθ-μcosθ×g×2t2.orsrough=sinθ-μcosθ×g×2t2...............(5)

Now, comparing equations (1) and (5) we get,

12gsinθ×t2=sinθ-μcosθ×g×2t2.orsinθ-μcosθ=sinθ4orsinθ-μcosθsinθ=14or1-μcotθ=14orμcotθ=34orμ=34cotθ=34tanθorμ=34tanθ

So, the coefficient of friction is μ=34tanθ.


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