A simple pendulum of length l and having a bob of mass M is suspended in a car.
The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
Now when the pendulum will be in a car moving on a circular track of radius R with a uniform speed v, the effective acceleration on it will change which was earlier only acceleration due to gravity and now there is acceleration due to gravity and centripetal acceleration both acting perpendicular to each other so net acceleration and its direction also will change, the mean position will also change and the new mean position or equilibrium position of bob of pendulum will be in direction of acceleration as shown in the figure
Now centripetal acceleration on a body moving in circular path with uniform speed is given as
ac = v2/R
where v is the uniform speed of the body and R is the radius of circular path
acceleration due to gravity g is acting in downward direction so net acceleration is
we know time period of oscillation of a pendulum is given as
Where T is the time period l is the length of the pendulum and g is acceleration due to gravity
Since here now effective acceleration acting on pendulum has changed the equation will become
Where a is the effective acceleration acting on bob of pendulum
Putting value of a we get time period of oscillation of pendulum
Where T is the time period l is the length of the pendulum which is in a car which is moving on a circular track of radius R with a uniform speed v and g is acceleration due to gravity.
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