Question

The period \(T\) of a simple pendulum is the amount of time required for it to undergo one complete oscillation. If the length of the pendulum is \(L\) and the acceleration due to gravity is \(g\), then the period is given by $$ T=2 \pi L^{p} g^{q} $$ Find the powers \(p\) and \(q\) required for dimensional consistency.

Short answer

The powers are \( p = \frac{1}{2} \) and \( q = -\frac{1}{2} \).

Step-by-step solution

Understand the formula components

The given formula for the period of a pendulum is \( T = 2\pi L^p g^q \). We aim to find the values of \( p \) and \( q \) so the formula is dimensionally consistent. Here, \( L \) is the length of the pendulum and has dimensions of length \([L]\), \( g \) is the acceleration due to gravity having dimensions of \([LT^{-2}]\), and \( T \) is the period with dimensions \([T]\).

Write dimensions for each component

Express the dimensions of each component:- The period \( T \) is \([T] = [T]\).- The length \( L \) is \([L] = [L]\).- The acceleration \( g \) is \([g] = [LT^{-2}]\).

Formulate the dimensional equation

Substitute the dimensions into the given pendulum period formula:\[[T] = [L]^p [LT^{-2}]^q.\]This equation should hold dimensionally, meaning both sides of the equation need to match in dimensions.

Simplify the dimensional equation

Apply the properties of exponents to simplify:- \( [L]^p \) becomes \( [L^p] \).- \( [LT^{-2}]^q \) becomes \( [L^q][T^{-2q}] \).Substitute into the dimensional equation:\[[T] = [L^{p+q}] [T^{-2q}].\]

Equate dimensions for consistency

For the equation to be dimensionally consistent, the powers of \( L \) and \( T \) must match on both sides:- Compare length dimensions: \([L^0] = [L^{p+q}] \Rightarrow p + q = 0\).- Compare time dimensions: \([T^1] = [T^{-2q}] \Rightarrow -2q = 1\).

Solve for p and q

From the time dimension equation, solve \(-2q = 1\) to get \( q = -\frac{1}{2} \).Using the value of \( q \) in the length dimension equation, substitute into \( p + q = 0 \) to get \( p - \frac{1}{2} = 0 \) which gives \( p = \frac{1}{2} \).

Key concepts

Pendulum Period Formula

The pendulum period formula gives us the time taken for a pendulum to swing back and forth for one complete cycle. It's essential for understanding how time and motion relate in pendulum systems. The formula is expressed as:\[ T = 2\pi L^{p} g^{q} \]where \( T \) is the period of the pendulum, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity, commonly approximated as \( 9.8 \text{ m/s}^2 \) on Earth's surface. This formula allows us to predict how changes in length \(L\) and gravitational pull \(g\) will affect the period \(T\).Understanding the pendulum period formula helps in predicting pendulum behavior in different environments, such as:

• Describing the simple harmonic motion in clocks.
• Predicting changes in environments with varying gravity.
• Observing how adjusting the pendulum’s length can change clock time intervals.

Pendulums are incredibly useful in timekeeping, physics experiments, and educational demonstrations of motion and force. Grasping how to calculate the period with this formula is a crucial physics skill.

Dimensional Consistency

Dimensional consistency means ensuring both sides of an equation have the same dimensional units. It is a crucial check in physics to confirm the validity of an equation.Ensuring the formula for the pendulum period is dimensionally consistent involves verifying each term within the formula has compatible dimensions. This process helps prevent errors arising from adding or equating terms with incompatible dimensions.For example, in our pendulum period formula \( T = 2\pi L^{p} g^{q} \):

• Ensure the dimensions of \( T \) on the left side are equal to the dimensions on the right side containing elements with \( L \) and \( g \).
• Express each component in terms of fundamental dimensions, here \( L \) for length, and \( T \) for time.

The goal is to simplify the exponents of each dimension, ensuring they add up to match the dimensions of the period \([T]\). Checking dimensional consistency helps identify the powers \( p \) and \( q \), which in this case are \( \frac{1}{2} \) and \(-\frac{1}{2} \) respectively, ensuring the formula remains valid. It is fundamental because consistent units help quantify the physical relationship between period, length, and gravitational acceleration.

Pendulum Oscillation

Pendulum oscillation refers to the repetitive movement a pendulum makes through a fixed arc determined by gravitational force and length. It's a classic example of simple harmonic motion, where the force restoring the pendulum to equilibrium is proportional to its displacement. The properties of pendulum oscillation include:

• The pendulum's period is independent of its total mass.
• It remains consistent for small angle swings (simple harmonic approximation).
• It exhibits periodic motion, which repeats in equal intervals of time.

Understanding pendulum oscillation is genuinely beneficial for multiple applications:

• Used in calibrating time in clocks, providing precision and reliability.
• Helps physicists understand gravitational fields by measuring time periods.
• Represents a perfect model for studying frictionless motion and energy conservation in systems.

The knowledge of pendulum oscillation enhances your comprehension of broader concepts like wave motion, resonance, and energy transformations in mechanical systems. Pendulums serve as a tangible means to visualize these fundamental concepts in physics.