3. 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, as shown in this graph.


Answer the following questions:
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.
Show any necessary calculations.

a. To find the period and frequency of the pendulum's motion, we need to use the formula:

Period (T) = 2π√(L/g)

Where:
L = length of the pendulum (0.200 m)
g = acceleration due to gravity

To find the frequency, we can use the formula:

Frequency (f) = 1 / Period

Now, to calculate the period and frequency, we need to determine the acceleration due to gravity on the surface of the planet.

b. To determine the phase difference between the displacements, velocity, and acceleration graphs, we need to understand the relationship between them.

The velocity (v) of the pendulum is given by the equation:

v = Aωsin(ωt + φ)

Where:
A = amplitude (maximum displacement)
ω = angular frequency (2πf)
t = time
φ = phase constant

The acceleration (a) of the pendulum is given by the equation:

a = Aω²cos(ωt + φ)

There is a phase difference of π/2 (90 degrees) between the displacement and velocity graphs and an additional phase difference of π/2 (90 degrees) between the displacement and acceleration graphs.

c. To find the acceleration due to gravity on the surface of the planet and convert it to g-forces, we need to use the relationship:

Acceleration due to gravity (g) = 4π²L/T²

Where:
L = length of the pendulum (0.200 m)
T = period (found in part a)

By plugging in the values, we can calculate the acceleration due to gravity on the surface of the planet.

Now, let's calculate the answers step by step.

a. Period and Frequency:
Using the formula: Period (T) = 2π√(L/g)

Step 1: Determine the value of g (acceleration due to gravity) on the planet's surface.
To find g, you may need to research or be given the information about the planet's properties.

Step 2: Calculate the period.
T = 2π√(L/g)

Step 3: Calculate the frequency.
Frequency (f) = 1 / Period

b. Phase difference between displacement, velocity, and acceleration:
There is a phase difference of π/2 (90 degrees) between the displacement and velocity graphs and an additional phase difference of π/2 (90 degrees) between the displacement and acceleration graphs.

c. Acceleration due to gravity and g-forces:
Use the formula: Acceleration due to gravity (g) = 4π²L/T²

Step 1: Calculate the acceleration due to gravity.
g = 4π²L/T²

Step 2: Convert the acceleration due to gravity to g-forces.
1 g-force is equivalent to the acceleration due to gravity on Earth (9.8 m/s²). Divide the acceleration due to gravity on the planet by 9.8 to get the number of g-forces.