Question

A car is moving with a constant speed of $20\, m / s$ in a circular horizontal track of radius $40\, m$ A bob is suspended from the roof of the car by a massless string The angle made by the string with the vertical will be : (Take $g=10 \, m / s ^2$ )

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{3}$

Answer 1

The Correct Option is C

To find the angle made by the string with the vertical, we need to consider the forces acting on the bob while the car is moving in a circular path.

1. The bob experiences two forces: the gravitational force acting downward, \(mg\), and the tension \(T\) in the string, which can be resolved into two components.
   • The vertical component, \(T \cos \theta\), balances the gravitational force \(mg\).
   • The horizontal component, \(T \sin \theta\), provides the centripetal force needed for circular motion.
2. Thus, we can write the equations:
   • \(T \cos \theta = mg\) (equation 1)
   • \(T \sin \theta = \frac{mv^2}{r}\) (equation 2), where \(v = 20 \, \text{m/s}\) and \(r = 40 \, \text{m}\).
3. Dividing equation 2 by equation 1 gives:
   • \(\tan \theta = \frac{v^2}{rg}\)
4. Substitute the given values:
   • \(\tan \theta = \frac{(20)^2}{40 \times 10} = \frac{400}{400} = 1\)
5. Solving for \(\theta\):
   • Since \(\tan \theta = 1\), we have \(\theta = \frac{\pi}{4}\) (or 45 degrees).

Thus, the angle made by the string with the vertical is \(\frac{\pi}{4}\). This matches the given correct option.