An astronaut lands on an alien planet. He places a pendulum (L = 0.200 m) on the surface and sets it in simple harmonic motion.
a. What is the period and frequency of the pendulum’s motion?
b. How many seconds out of phase with the displacements shown would graphs of the velocity and acceleration be?
c. What is the acceleration due to gravity on the surface of the planet in m/s2? Determine the number of g-forces.
To answer these questions, we need to understand the basic principles of a pendulum and how they relate to the given information.
a. Period and Frequency of the Pendulum's Motion:
- The period (T) of a pendulum is the time it takes to complete one full oscillation, back and forth.
- The frequency (f) of a pendulum is the number of complete oscillations it makes per second.
The formula to calculate the period of a pendulum is:
T = 2π √(L/g)
where:
- T is the period in seconds
- π is a mathematical constant (approximately 3.14159)
- L is the length of the pendulum in meters
- g is the acceleration due to gravity in m/s²
Using the given information L = 0.200 m, we need to determine the acceleration due to gravity on the alien planet.
b. Phase Difference Between Displacement, Velocity, and Acceleration:
The phase difference between the displacement, velocity, and acceleration of a simple harmonic motion is π/2 (or 90 degrees).
This means that when the displacement of the pendulum is at its maximum, the velocity is zero, and when the displacement is zero (at the center), the velocity is maximum in the opposite direction.
Similarly, the acceleration is maximum at the maximum displacement and zero at the center.
c. Acceleration due to Gravity on the Alien Planet:
To calculate the acceleration due to gravity (g) on the surface of the planet, we can use the formula:
g = (4π²L) / T²
where:
- g is the acceleration due to gravity in m/s²
- L is the length of the pendulum in meters
- T is the period in seconds
Using the values L = 0.200 m and T obtained from part a, we can determine the acceleration due to gravity on the alien planet.
To calculate the number of g-forces, we divide the acceleration due to gravity on the alien planet by the acceleration due to gravity on Earth (approximately 9.8 m/s²).
Now let's calculate the answers step by step.
a. Period and Frequency:
Using the formula T = 2π √(L/g) and the given value L = 0.200 m, we need to determine the value of g on the alien planet.
b. Phase Difference:
The phase difference between displacement, velocity, and acceleration in simple harmonic motion is π/2.
c. Acceleration due to Gravity:
Using the formula g = (4π²L) / T² and the obtained value of T from part a, we can calculate the acceleration due to gravity on the alien planet.
Once we have the acceleration due to gravity, we can determine the number of g-forces by dividing it by 9.8 m/s².
Let's proceed with the calculations.