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Question
Derive an expression for maximum speed moving along a horizontal circular track.
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Solution
When an object moves along a horizontal circular track, the force providing the necessary centripetal force is the frictional force between the object and the surface. The object will move with maximum speed when the frictional force is at its maximum value, which is given by limiting friction.
Consider an object of mass m moving on a horizontal circular track of radius r. The forces acting on the object are:
- Normal Reaction Force: N exerted by the surface (acts vertically upward).
- Gravitational Force: mg (acts vertically downward).
- Frictional Force: f (acts towards the center of the circular path to provide the required centripetal force).
Since the object is moving on a horizontal surface, the normal reaction is equal to the weight of the object:
N = mg
For circular motion, the centripetal force required is:
`Fc = (mv_"max"^2)/r`
Here, vmax is the maximum speed of the object.
The maximum frictional force available to provide this centripetal force is given by:
fmax = μN = μmg
where μ is the coefficient of friction between the object and the surface.
At maximum speed, the entire frictional force provides the necessary centripetal force:
`(mv_"max"^2)/r = μmg`
Canceling m from both sides:
`v_"max"^2/r = μg`
`v_max = sqrt(μgr)`
This formula indicates that the maximum speed is determined by the track's radius, the gravitational force, and the friction between the track and the object.
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