When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.


b. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain.

Verified step by step guidance

1. First, let's understand the relationship between the period T of a pendulum and the acceleration due to gravity g. The formula given is T = 2π(L/g)¹/², where L is the length of the pendulum.
2. Notice that T is proportional to the square root of the inverse of g, which means T = 2π√(L/g). This implies that as g increases, the value of √(L/g) decreases because g is in the denominator.
3. Since T is directly proportional to √(L/g), if g increases, √(L/g) decreases, leading to a decrease in T. Therefore, the period T of the pendulum decreases as g increases.
4. A decrease in the period T means that the pendulum completes its oscillations more quickly. Thus, the pendulum clock will speed up as g increases.
5. In summary, if the acceleration due to gravity g increases, the period T of the pendulum decreases, causing the pendulum clock to speed up.

Key Concepts

Pendulum Period Formula

The period T of a pendulum is given by the formula T = 2π(L/g)¹/², where L is the length of the pendulum and g is the acceleration due to gravity. This formula shows that the period is directly related to the square root of the length and inversely related to the square root of gravity. Understanding this relationship is crucial for analyzing how changes in g affect T.

Effect of Gravity on Pendulum

Gravity affects the pendulum's period; as gravity increases, the period decreases, meaning the pendulum swings faster. Conversely, if gravity decreases, the period increases, and the pendulum swings slower. This inverse relationship is essential for predicting how a pendulum clock's speed changes with variations in gravity.

Pendulum Clock Speed

A pendulum clock's speed is determined by the pendulum's period. If the period decreases due to an increase in gravity, the clock will speed up, as it completes more cycles in a given time. Conversely, if the period increases, the clock will slow down. Understanding this concept helps in determining the clock's behavior when moved to different locations with varying gravity.