A certain body remains stationary on the vertical inner wall of a cylindrical drum of radius 'r', rotating at a constant speed. Show that the minimum angular speed of the drum is where "u" is coefficient of friction g Vur between the body and surface of the wall.
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Answer:
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Step-by-step explanation:
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The coefficient of friction must be equal to the acceleration due to gravity. u = g
Given,
radius = r
coefficient of friction = u
To find,
The minimum angular speed of the drum is wherever "u" is constant coefficient of friction g Vur between the body and surface of the wall.
Solution,
The minimum angular speed of the drum can be determined by considering the forces acting on the body. The gravitational force acts downward on the body, while the normal force from the drum's wall acts perpendicular to the wall. The friction force acts parallel to the wall in the direction opposite to the motion of the body.
In order for the body to remain stationary on the wall, the net force on the body must be zero. This means that the gravitational force must be balanced by the normal and friction forces:
Fg + Fn + Ff = 0
Where Fg is the gravitational force, Fn is the normal force, and Ff is the friction force.
The gravitational force can be expressed as the mass of the body times the acceleration due to gravity:
Fg = mg
The normal force is equal in magnitude to the gravitational force, but acts in the opposite direction:
Fn = -mg
The friction force is equal in magnitude to the coefficient of friction times the normal force:
Ff = u * Fn
Substituting these expressions into the equation for the net force, we get:
mg - mg + u * (-mg) = 0
Solving for the coefficient of friction, we find that the minimum angular speed of the drum is:
u = g
If the coefficient of friction is less than this value, the body will slide down the wall of the drum. If the coefficient of friction is greater than this value, the body will remain stationary on the wall.
This means that in order for the body to remain stationary on the wall of the rotating drum, the coefficient of friction must be equal to the acceleration due to gravity. u = g
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