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 ?
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 ?
The bob of the simple pendulum will experience the acceleration due to gravity and the centripetal acceleration provided by the circular motion of the car.
Acceleration due to gravity $=g$
Centripetal acceleration $=\frac{v^{2}}{R}$
Where, $v$ is the uniform speed
of the car
$R$ is the radius of the track
Effective acceleration ( $\left.a_{\text {eff }}\right)$ is given as:
$a_{ eff }=\sqrt{g^{2}+\left(\frac{v^{2}}{R}\right)^{2}}$
Time period, $T=2 \pi \sqrt{\frac{l}{a_{\text {eff }}}}$
Where, $l$ is the length of the pendulum
Time period $T=2 \pi\sqrt{\frac{1}{g+\frac{v^{2}}{R^{2}}}}$
Similar Questions
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- [AIEEE 2012]
In the following table relation of graph in column$-I$ and shape of graph in column$-II$ is shown match them appropriately.
column$-I$
column $-II$
$(a)$ ${T^2} \to l$
$(i)$ Linear
$(b)$ ${T^2} \to g$
$(ii)$ Parabolic
$(c)$ ${T} \to l$
$(iii)$ Hyperbolic
In the following table relation of graph in column$-I$ and shape of graph in column$-II$ is shown match them appropriately.
| column$-I$ | column $-II$ |
| $(a)$ ${T^2} \to l$ | $(i)$ Linear |
| $(b)$ ${T^2} \to g$ | $(ii)$ Parabolic |
| $(c)$ ${T} \to l$ | $(iii)$ Hyperbolic |
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