Question

An 85.0-kg mountain climber plans to swing down, starting from rest, from a ledge using a light rope 6.50 m long. He holds one end of the rope, and the other end is tied higher up on a rock face. Since the ledge is not very far from the rock face, the rope makes a small angle with the vertical. At the lowest point of his swing, he plans to let go and drop a short distance to the ground. (a) How long after he begins his swing will the climber first reach his lowest point? (b) If he missed the first chance to drop off, how long after first beginning his swing will the climber reach his lowest point for the second time?

Short answer

(a) 2.56 seconds; (b) 5.11 seconds.

Step-by-step solution

Modeling the Pendulum Swing

We can model the mountain climber as a simple pendulum since he swings from a fixed point. The time period of a simple pendulum is given by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length of the rope (6.50 m) and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²).

Calculate Time Period for One Complete Swing

Substitute the known values into the formula: \( T = 2\pi \sqrt{\frac{6.50}{9.81}} \). Calculate \( \sqrt{\frac{6.50}{9.81}} \) to get approximately 0.814. Therefore, \( T = 2\pi \times 0.814 \), which is approximately 5.11 seconds.

Determine Time to Reach Lowest Point Once

The climber reaches the lowest point once when the pendulum completes half a period. Therefore, the time to first reach the lowest point is \( \frac{T}{2} = \frac{5.11}{2} \), which equals approximately 2.56 seconds.

Determine Time to Reach Lowest Point Again

For the second time the climber reaches the lowest point, the pendulum must complete one full period. Since one full period is already calculated to be 5.11 seconds, the climber reaches the lowest point for the second time at 5.11 seconds after beginning the swing.

Key concepts

Pendulum Period

The period of a pendulum is the time it takes for the pendulum to complete one full swing, from its starting point, down to the lowest point, back to the starting point, and returning to its lowest point again. The formula for calculating the period of a simple pendulum is given by:

• \( T = 2\pi \sqrt{\frac{L}{g}} \)

Here, \( T \) is the period, \( L \) is the length of the pendulum (or rope, in our scenario), and \( g \) is the gravitational acceleration. To solve the problem from the exercise, we use a rope length \( L = 6.50 \text{ m} \) and gravitational acceleration \( g \approx 9.81 \text{ m/s}^2 \).
This results in \( T = 2\pi \sqrt{\frac{6.50}{9.81}} \approx 5.11 \text{ seconds} \). This tells us that it takes about 5.11 seconds for the climber to complete a full swing cycle.

Pendulum Motion

Pendulum motion is a kind of periodic motion seen in objects that swing back and forth from a fixed point. Such motion is characterized by a repetitive pattern where the object reaches its peak angular excursion on one side, swings through its lowest point, and then reaches the same peak on the opposite side.
In this exercise, the mountain climber swings like a pendulum. As the rope makes a small angle with the vertical, it approximates simple pendulum motion. Each swing is predictable so that crucial timings, like reaching the lowest point, can be calculated.

• The first time he reaches the lowest point occurs halfway through his swing cycle, \( \frac{T}{2} = 2.56 \text{ seconds} \).
• If he misses this opportunity, he will reach the lowest point again at full period completion, at \( 5.11 \text{ seconds} \).

Gravitational Acceleration

Gravitational acceleration \( g \) is the acceleration due to gravity experienced by objects in free fall. On Earth, this value is approximately \( 9.81 \text{ m/s}^2 \). This acceleration is crucial in determining the motion of pendulums.
It influences how fast or slow a pendulum swings. A heavier gravitational force would result in a shorter period, meaning the pendulum swings back and forth more quickly. Conversely, a weaker gravitational force would result in a longer period.
Understanding gravitational acceleration helps in accurately using the pendulum period formula \( T = 2\pi \sqrt{\frac{L}{g}} \), as the required period greatly depends on accurate estimation of \( g \). When solving pendulum problems, assuming gravity remains constant at \( 9.81 \text{ m/s}^2 \) simplifies calculations and allows for accurate predictions about pendulum movement.