Problem 56 - College Physics

Question

\(\bullet\) \(\bullet\) An astronaut notices that a pendulum which took 2.50 s for a complete cycle of swing when the rocket was waiting on the launch pad takes 1.25 s for the same cycle of swing during liftoff. What is the acceleration of the rocket? (Hint: Inside the rocket, it appears that \(g\) has increased.)

Step-by-Step Solution

Answer

The rocket's acceleration is 29.43 m/s².

Step 1: Understand the Pendulum Period Formula

The period of a pendulum is given by the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity. During the rocket's liftoff, the effective acceleration due to gravity changes.

Step 2: Calculate Initial Conditions

Initially, the pendulum period is \( T_1 = 2.50 \) seconds on the launch pad where the gravitational acceleration is \( g = 9.81 \, \text{m/s}^2 \). Substitute into the formula: \( 2.50 = 2\pi \sqrt{\frac{L}{9.81}} \). Square both sides to solve for \( L \): \( L = \frac{(2.50)^2 \times 9.81}{4\pi^2} \).

Step 3: Calculate Conditions During Liftoff

During liftoff, the period is \( T_2 = 1.25 \) seconds. Substitute into the period formula: \( 1.25 = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \). Square both sides to solve for \( g_{\text{effective}} \): \( g_{\text{effective}} = \frac{L \times 4\pi^2}{1.25^2} \).

Step 4: Substitute and Solve for Effective Gravity

Substitute the expression for \( L \) from Step 2 into the equation for \( g_{\text{effective}} \) from Step 3: \( g_{\text{effective}} = \frac{(\frac{(2.50)^2 \times 9.81}{4\pi^2}) \times 4\pi^2}{1.25^2} \). Simplify to find \( g_{\text{effective}} = (\frac{2.50}{1.25})^2 \times 9.81 = 4 \times 9.81 = 39.24 \text{ m/s}^2 \).

Step 5: Calculate the Rocket's Acceleration

The effective gravity during liftoff \( g_{\text{effective}} \) is the sum of Earth's gravity and the rocket's acceleration: \( g_{\text{effective}} = g + a \). Thus, \( a = g_{\text{effective}} - g = 39.24 - 9.81 \).

Step 6: Final Calculation

Substitute the known values to find the rocket's acceleration: \( a = 39.24 - 9.81 = 29.43 \text{ m/s}^2 \).

Key Concepts

Acceleration Due to Gravity

Gravity is an invisible force that pulls objects towards the center of the Earth. This force is why apples fall to the ground and pendulums swing back and forth. The acceleration due to gravity, represented by the letter \( g \), is approximately \( 9.81 \, \text{m/s}^2 \) on the surface of the Earth. This value can slightly vary depending on where you are on the planet, whether you're standing at sea level or climbing a mountain. But in general, it's a constant figure used in many physics calculations.

Gravity not only keeps us grounded but also affects the motion of objects, such as a pendulum. When a pendulum swings, gravity plays a critical role by pulling it downwards, which contributes to its back and forth motion.

Pendulum Formula

To understand how a pendulum moves, there is a special formula that connects the length of the pendulum and gravity to its period (the time it takes to complete a full swing). The formula is: \[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity. This formula helps us figure out how long it takes for a pendulum to swing back and forth.

In the exercise mentioned, an astronaut uses this formula to determine changes in his pendulum's period when the rocket is at rest and during liftoff. When the gravitational force changes, like when a rocket accelerates upwards, the period of the pendulum changes as well. This is a great example of how physics formulas relate to real-world scenarios!

Effective Gravity

Effective gravity is the gravitational effect experienced inside a moving environment, like a rocket. It's like being in an elevator. When the elevator goes up, you feel heavier, and when it goes down, you might feel lighter. This is because effective gravity combines Earth's gravity with any other forces, such as the additional force from the acceleration of the rocket.

Effective gravity can be calculated by adding the usual gravitational acceleration to the acceleration of the moving platform, like a rocket:
\[ g_{\text{effective}} = g + a \]
In the given problem, effective gravity was increased during the liftoff because the rocket accelerates upwards, adding to the standard gravitational pull.

Rocket Acceleration

During a rocket's launch, it undergoes acceleration, which means its velocity is increasing as it moves upwards. This acceleration is why astronauts feel heavier during liftoff.

When analyzing a pendulum inside the rocket, its period decreased because the effective gravity increased due to the rocket's acceleration. By using the pendulum's changed period and applying the pendulum formula, we can calculate the rocket's acceleration. This involves finding the difference between effective gravity and Earth's gravity:

• Using \( g_{\text{effective}} = g + a \)
• Solving for \( a \) gives \( a = g_{\text{effective}} - g \).

In the astronaut's case, the rocket's acceleration was found to be \( 29.43 \, \text{m/s}^2 \), highlighting how physics can be practical and revealing when looking at detailed phenomena.