Problem 87 - University Physics with Modern Physics

Question

CP On the planet Newtonia, a simple pendulum having a bob with mass 1.25 \(\mathrm{kg}\) and a length of 185.0 \(\mathrm{cm}\) takes 1.42 \(\mathrm{s}\) . when released from rest, to swing through an angle of \(12.5^{\circ}\) , where it again has zero speed. The circumference of Newtonia is measured to be \(51,400 \mathrm{km}\) . What is the mass of the planet Newtonia?

Step-by-Step Solution

Answer

The mass of Newtonia is approximately \(9.56 \times 10^{26}\) kg.

Step 1: Understand the Pendulum

In this problem, we have a pendulum with a bob mass of 1.25 kg, a length of 185.0 cm, and it takes 1.42 seconds to travel through an angle of 12.5 degrees and back to zero speed. The pendulum exhibits simple harmonic motion. This will help us find the local gravitational acceleration (g).

Step 2: Convert Units

Convert the length of the pendulum from centimeters to meters. The length in meters is 185.0 cm which equals 1.85 meters.

Step 3: Use Harmonic Motion Formula

Use the formula for the period of a simple pendulum: \( T = 2\pi \sqrt{\frac{L}{g}} \), where \(T = 1.42\) seconds and \(L = 1.85\) meters. Solve for \(g\).

Step 4: Solve for Gravitational Acceleration

Rearrange the pendulum formula to \(g = \frac{4\pi^2 L}{T^2}\). Substitute \(L = 1.85\) m and \(T = 1.42\) seconds into the equation to find \(g\): \[ g = \frac{4 \times \pi^2 \times 1.85}{(1.42)^2} \approx 36.2 \text{ m/s}^2 \].

Step 5: Understand Circumference and Gravity Relationship

Using the circumference of the planet \(C = 51,400 \text{ km} = 51,400,000 \text{ meters}\), use the relationship between gravity \(g\) and the radius \(R\) of a planet: \( g = \frac{GM}{R^2} \) with \(C = 2\pi R\).

Step 6: Calculate Radius from Circumference

Find the radius \(R\) of Newtonia using \(R = \frac{C}{2\pi}\). So, \(R = \frac{51,400,000}{2\pi} \approx 8,184,115 \text{ m} \).

Step 7: Solve for Mass of Newtonia

Using \(g = \frac{GM}{R^2}\), rearrange to find \(M = \frac{gR^2}{G}\), where \(G = 6.674 \times 10^{-11} \text{ m}^3 \text{/kg/s}^2\). Substitute \(g = 36.2 \text{ m/s}^2\) and \(R = 8,184,115 \text{ m}\): \[ M = \frac{36.2 \times (8,184,115)^2}{6.674 \times 10^{-11}} \approx 9.56 \times 10^{26} \text{ kg} \].

Key Concepts

Simple Harmonic Motion

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. An object moves back and forth over the same path. A classic example of this is a pendulum swinging from side to side.
When dealing with pendulums, the period is the time it takes for one complete cycle of motion. This can be calculated using the formula for a simple pendulum:

• \( T = 2\pi \sqrt{\frac{L}{g}} \)

Here, \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.
Despite changes in the amplitude, the period remains constant if the oscillations are small, which makes it suitable to model scenarios like the one described in the Newtonia problem.
To solve the original problem, inverse the function to find \( g \) given \( T \) and \( L \). This involves rearranging our simple harmonic motion formula to isolate \( g \) and then plugging in the known values.

Gravitational Acceleration

Gravitational acceleration is crucial in measuring how fast objects fall towards a planet due to its gravity. In simple terms, it's the change in velocity an object experiences as it's pulled by the planet's force of gravity.
On Earth's surface, \( g \) is approximately 9.81 \( \text{m/s}^2 \), but on other planets, the value of \( g \) can vary. This variation relies on factors like the planet's mass and radius.

• The formula to determine gravitational acceleration is: \[ g = \frac{4\pi^2 L}{T^2} \]

Here, \( L \) is the pendulum's length and \( T \) the period.
In the Newtonia problem, we calculated \( g \) as 36.2 \( \text{m/s}^2 \), indicating a stronger force of gravity than on Earth. This will give insights into the planet’s mass, and its size compared to the forces we're familiar with.

Planetary Mass Calculation

Calculating a planet's mass involves knowing its gravitational influence and dimensions. Using this, scientists derive the planet's mass to understand its make-up and behavior in space.
The formula stems from Newton's law of gravitation, which ties together the gravitational force, the mass of the objects involved, the distance between their centers, and the gravitational constant \( G \).

• Given: \[ g = \frac{GM}{R^2} \]

Where \( g \) is gravitational acceleration, \( M \) the planet's mass, \( R \) its radius, and \( G \) the universal gravitational constant, \( 6.674 \times 10^{-11} \text{ m}^3 \text{/kg/s}^2 \).
For Newtonia, the process involves determining \( R \) using its circumference (\( C = 51,400,000 \text{ meters} \)), derived from \( R = \frac{C}{2\pi} \). Then plug this \( R \) and previously found \( g \) into the mass formula. Doing so reveals Newtonia’s mass to be approximately \( 9.56 \times 10^{26} \text{ kg} \), showcasing its hefty size and gravitational strength compared to familiar planets.