Question

6 formula 1 point a pendulum bob is attached to a very light string. the string length l is equal to 11 meters. the initial angle that the pendulum bob is released at is 30 degrees from the vertical (not quite drawn to scale in the diagram). calculate the maximum speed of the pendulum. assume there is no air resistance. \\(\theta\\) = angle relative to pendulum at rest answer previous next

Explanation:

Step1: Determine the height difference

The height difference \( h \) the pendulum bob falls can be found using the string length \( L \) and the angle \( \theta \). The vertical drop is \( h = L(1 - \cos\theta) \). Given \( L = 11 \, \text{m} \) and \( \theta = 30^\circ \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \). So \( h = 11(1 - 0.866) = 11 \times 0.134 \approx 1.474 \, \text{m} \).

Step2: Apply conservation of energy

At the highest point, the pendulum has gravitational potential energy \( mgh \), and at the lowest point (maximum speed), all that potential energy is converted to kinetic energy \( \frac{1}{2}mv^2 \). Setting them equal: \( mgh = \frac{1}{2}mv^2 \). The mass \( m \) cancels out, so \( v = \sqrt{2gh} \).

Step3: Calculate the speed

Substitute \( g = 9.8 \, \text{m/s}^2 \) and \( h \approx 1.474 \, \text{m} \) into the formula: \( v = \sqrt{2 \times 9.8 \times 1.474} \approx \sqrt{28.99} \approx 5.38 \, \text{m/s} \).

Answer:

\( \approx 5.38 \, \text{m/s} \)