Question

A horizontal tree branch is directly above another horizontal tree branch. The elevation of the higher branch is \(9.65 \mathrm{~m}\) above the ground, and the elevation of the lower branch is \(5.99 \mathrm{~m}\) above the ground. Some children decide to use the two branches to hold a tire swing. One end of the tire swing's rope is tied to the higher tree branch so that the bottom of the tire swing is \(0.470 \mathrm{~m}\) above the ground. This swing is thus a restricted pendulum. Starting with the complete length of the rope at an initial angle of \(14.2^{\circ}\) with respect to the vertical, how long does it take a child of mass \(29.9 \mathrm{~kg}\) to complete one swing back and forth?

Short answer

Answer: The time it takes for the child to complete one back-and-forth swing on the tire swing is approximately 6.05 seconds.

Step-by-step solution

Calculate the length of the rope

We are given 3 values regarding the heights. The height of the higher tree branch (9.65 m), the height of the lower tree branch (5.99 m), and the height of the tire swing above the ground (0.47) when at its lowest position. We can find the length of the rope (L) by subtracting 0.47 from the height of the higher tree branch: L = 9.65 m - 0.47 m = 9.18 m

Calculate the period of the pendulum

The period of a pendulum (the time it takes to complete one back-and-forth swing) can be calculated using the formula: T = 2π √(L/g) Where T is the period, L is the length of the rope (9.18 m in this case), and g is the acceleration due to gravity (approximately 9.81 m/s²). So, we can plug in the values: T = 2π √((9.18 m)/(9.81 m/s²))

Solve for the period T

Now it's time to solve for the period. When calculation with the given values, we find: T = 2π √((9.18 m)/(9.81 m/s²)) ≈ 6.05 s

Write down the final result

Now we have the time it takes for the child of mass 29.9 kg to complete one swing (back and forth) on the tire swing, which is approximately: T ≈ 6.05 s

Key concepts

Simple Pendulum

Imagine a weight suspended from a fixed point with a string or rod that is allowed to swing freely back and forth under the influence of gravity. This is a simple pendulum. It's a classic example of an oscillatory system and has been instrumental in our understanding of timekeeping and physics in general.

When discussing a simple pendulum, there are several key factors to consider. Its length, defined as the distance from the pivot point to the center of mass of the pendulum bob, is the most critical variable because it directly affects the pendulum's period. Other elements include the bob's mass, although it surprisingly does not impact the period, and the acceleration due to gravity, which is a constant that affects all gravitational phenomena on Earth. The motion of a simple pendulum is predictable and periodic, making it a perfect candidate to delve into the concepts of harmonic motion and calculating the period of oscillation.

Period of a Pendulum

The period of a pendulum is the time it takes for the pendulum to swing from its initial position to the furthest point in its trajectory and back again to the start. It is solely determined by the length of the pendulum and the acceleration due to gravity, an insight first discovered by Galileo Galilei.

Period Calculation Example

Using the exercise provided as reference, the period (\(T\)) can be calculated with the formula \(T = 2\pi \sqrt{L/g}\), where \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity. This formula is a key concept in understanding how pendulums behave. Even in the exercise where the pendulum (the tire swing) is set into motion at a slight angle, the period remains a constant for small angles, making the approximation valid and simplifying the swing into harmonic motion.

Harmonic Motion

Harmonic motion, or more specifically simple harmonic motion, is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It's exemplified perfectly by the swinging of a simple pendulum, where gravity provides the restoring force.

What makes this concept critical for understanding pendulums is that the motion is predictable and regular. For small oscillations, a pendulum's motion can be approximated to simple harmonic motion, which allows us to use the formula for the period mentioned earlier effectively. This model works under the assumption that the amplitude is small and the pendulum doesn't lose energy over time, thus no air resistance or friction at the pivot is considered. Real-world pendulums, like the tire swing in our exercise, often experience dampened oscillations over time, but for the purpose of calculating a single period, the simple harmonic motion model serves as an excellent approximation.