Question

A car moving with a speed of 54 km/h takes a turn of radius 20 m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take \(g = 10\) m/s²)

• A. \(\tan^{-1}(0.5) \)
• B. \(\tan^{-1}(0.75) \)
• C. \(\tan^{-1}(1.125) \)
• D. \(\tan^{-1}(0.25) \)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When a car turns, objects inside experience a centrifugal force. The pendulum bob will move outward until the horizontal component of the string tension balances the centrifugal force, and the vertical component balances gravity.
Step 2: Key Formula or Approach:
1. Convert speed to m/s: \(v (\text{m/s}) = v (\text{km/h}) \times \frac{5}{18}\).
2. Angle with vertical: \(\tan \theta = \frac{v^2}{rg}\).
Step 3: Detailed Explanation:
First, convert the speed: \[ v = 54 \times \frac{5}{18} = 3 \times 5 = 15 \text{ m/s} \] Given radius \(r = 20 \text{ m}\) and \(g = 10 \text{ m/s}^2\). Using the formula for the angle in a non-inertial frame: \[ \tan \theta = \frac{a_c}{g} = \frac{v^2}{rg} \] \[ \tan \theta = \frac{(15)^2}{20 \times 10} \] \[ \tan \theta = \frac{225}{200} = \frac{9}{8} = 1.125 \] \[ \theta = \tan^{-1}(1.125) \]
Step 4: Final Answer:
The angle made by the string with the vertical is \(\tan^{-1}(1.125)\).