A small body f mass m can slide without friction along a trough which ...
A small body f mass m can slide without friction along a trough which ...
Analysis:
To solve this problem, we can consider the forces acting on the body at rest in the trough. There are two forces to consider: the gravitational force and the centrifugal force.
Gravitational Force:
The gravitational force acting on the body is given by Fg = mg, where m is the mass of the body and g is the acceleration due to gravity.
Centrifugal Force:
The centrifugal force is the apparent outward force experienced by an object moving in a circular path. It can be calculated using the equation Fc = mw^2R, where m is the mass of the body, w is the angular velocity of the trough, and R is the radius of the trough.
Equilibrium Condition:
Since the body is at rest, the net force acting on it must be zero. Therefore, the gravitational force and the centrifugal force must balance each other.
Equating Forces:
We can equate the gravitational force and the centrifugal force to find the height at which the body will be at rest.
mg = mw^2R
Simplifying and Solving for height:
Dividing both sides of the equation by m, we get:
g = w^2R
Solving for R, we get:
R = g/w^2
The height h can be obtained by subtracting R from the radius of the trough:
h = R - R*cos(theta)
Final Answer:
The height at which the body will be at rest with respect to the trough is given by:
h = g/w^2 - (g/w^2)*cos(theta)
where g is the acceleration due to gravity, w is the angular velocity of the trough, and theta is the angle between the vertical axis and the line connecting the center of the trough to the body.
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