Question

Two children are on adjacent playground swings with chains of the same length. They are given pushes by an adult and then left to swing. Assuming that each child on a swing can be treated as a simple pendulum and that friction is negligible, which child takes the longer time for one complete swing (has a longer period)? a) the bigger child b) the lighter child c) neither child d) the child given the bigger push

Short answer

Answer: c) Neither child will have a longer period for a complete swing.

Step-by-step solution

Recall the period formula for a simple pendulum

The formula for the period of a simple pendulum is given by: T = 2π √(L/g) where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

Analyze the factors affecting the period

From the formula, we can see that the period of a simple pendulum depends only on the length (L) of the pendulum and the acceleration due to gravity (g). The mass or size of the child and the size of the push given do not affect the period of a simple pendulum.

Compare the periods of both children

Since both children are given swings with chains of the same length (L) and are affected by the same acceleration due to gravity (g), their periods will be the same, as T = 2π √(L/g) remains constant for both children. The answer is c) Neither child will have a longer period for a complete swing.

Key concepts

Pendulum Motion

The motion of a pendulum is a classic example of periodic motion observed in everyday life, such as in playground swings. A pendulum swings back and forth through a small arc, driven by gravitational forces, and exhibits simple harmonic motion under the simplifying assumption of small angles.

When a child on a swing is initially pushed, it gains kinetic energy that is gradually turned into potential energy as they reach the maximum height at either end of the swing's path. In the absence of friction and air resistance, this conversion between potential and kinetic energy would theoretically continue indefinitely, with the child swinging back and forth at a consistent rate and magnitude.

Pendulum motion can be disrupted by external factors like air resistance and friction or if the swing is pushed continuously. However, in a simplified scenario where these factors are negligible, the pendulum's motion is primarily governed by the effects of gravity and the length of the swing's chains.

Oscillation Period

The oscillation period of a pendulum, often represented by the symbol T, is the time it takes to complete one full cycle of motion, i.e., to swing from the starting point to the furthest extent of its path and back again. The period formula for a simple pendulum is beautifully succinct: \[ T = 2\pi \sqrt{\frac{L}{g}} \

The formula indicates that the oscillation period (T) is directly proportional to the square root of the length (L) of the pendulum and inversely proportional to the square root of the acceleration due to gravity (g), which is a constant. Therefore, if a pendulum's length is doubled, as per the formula, the period increases by a factor of \(\sqrt{2}\), whereas the mass of the pendulum and the initial push (provided it's within the small angle assumption for simple harmonic motion) do not affect the period.

Understanding the relationship between the period and the length of the pendulum leads to the counterintuitive realization that a heavier or lighter child on the swing, or how hard they are pushed initially, doesn't change the time it takes for a complete swing, assuming the swing's chain length remains unchanged.

Theoretical Mechanics

The study of pendulum motion falls within the realm of theoretical mechanics, a branch of physics that deals with the motion of objects and the forces that cause this motion. Theoretical mechanics can be divided into dynamics, which studies the forces and their effects, and kinematics, which studies the motion without considering forces.

The simple pendulum problem is approached through dynamics by considering the force of gravity and ignoring drag forces and friction that would usually cause the motion to damp out in a real-world scenario. By adopting the constraints of an ideal simple pendulum, theoretical mechanics simplifies the variables involved to focus purely on the fundamental principles governing the object's motion.

The formula for the period of a simple pendulum highlights the essence of theoretical mechanics: it provides a simplified model that helps us understand the underlying principles of physical phenomena, refining our comprehension of concepts like periodic motion and the influence of gravitational forces on bodies. This simplified approach, while not capturing every variable encountered in reality, allows us to make precise predictions about the behavior of pendulums in a controlled environment.