A simple pendulum of the length 0.350 m starts from rest at a max displacement of 10 degrees. from equilibrium position.

a) At what time will the pendulum be located at an angle of displacement of 8 degrees?
b) When will the pendulum return to its starting position?

To solve these questions, we need to use the equation of motion for a simple pendulum. The equation is given by:

T = 2π√(L/g)

where:
T is the period of the pendulum (time taken for one full swing)
L is the length of the pendulum
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Let's solve each question one by one:

a) At what time will the pendulum be located at an angle of displacement of 8 degrees?

To find the time it takes for the pendulum to reach a specific angle, we need to determine the number of periods it takes to reach that angle.

Step 1: Convert the angle to radians.
Angle in radians = (angle in degrees) * (π/180)

For the given problem, we have an angle of 8 degrees. Converting this to radians:
Angle in radians = 8 * (π/180) ≈ 0.139 radians

Step 2: Calculate the period of the pendulum using the given length.
Length of the pendulum, L = 0.350 m
Acceleration due to gravity, g = 9.8 m/s^2

T = 2π√(L/g)
T = 2π√(0.350/9.8)
T ≈ 1.118 seconds

Step 3: Calculate the time it takes to reach the desired angle.
The time taken to reach an angle is given by:
Time = (Number of periods) * (Period)

Since the pendulum starts from rest, it will first swing to the left and then return to the right. Each complete swing is one period.

Number of periods = (angle in radians) / (2π)
Number of periods = 0.139 / (2π)

Time = (0.139 / (2π)) * (1.118)
Time ≈ 0.196 seconds

b) When will the pendulum return to its starting position?

To find the time it takes for the pendulum to return to its starting position, we need to find the time for one complete swing, which is the period of the pendulum.

Using the same formula as before:
T = 2π√(L/g)
T = 2π√(0.350/9.8)
T ≈ 1.118 seconds

Therefore, the pendulum will return to its starting position after approximately 1.118 seconds.

Keep in mind that these calculations assume ideal conditions without any external forces or damping effects on the pendulum.

To solve these questions, we can use the equations of motion for a simple pendulum. The period of a simple pendulum is given by the formula:

T = 2π√(L/g)

Where:
T is the period of the pendulum
L is the length of the pendulum
g is the acceleration due to gravity (approximately 9.8 m/s^2)

a) At what time will the pendulum be located at an angle of displacement of 8 degrees?

To find the time at which the pendulum is located at an angle of displacement of 8 degrees, we can use the formula for the angular displacement of a simple pendulum:

θ = θ0 * cos(ωt)

Where:
θ is the angular displacement at a given time t
θ0 is the initial angular displacement (10 degrees in this case)
ω is the angular frequency of the pendulum (ω = 2π/T)

First, let's convert the angle from degrees to radians:
θ0 = 10 degrees * (π/180) = 10π/180 radians
θ = 8 degrees * (π/180) = 8π/180 radians

Next, we can solve for the time t using the formula for angular displacement:

8π/180 = 10π/180 * cos(ωt)

Now, rearrange the equation to solve for t:

cos(ωt) = (8π/180) / (10π/180)

We know that cos(θ) = adjacent/hypotenuse, so in this case:

cos(ωt) = (8π/180) / (10π/180) = 8/10 = 0.8

Using the inverse cosine function (cos⁻¹), we can find the value of ωt:

ωt = cos⁻¹(0.8)

Using a calculator, we find that ωt = 0.6435 radians.

Now, we can calculate the time t using the formula:

ω = 2π/T

Substituting ωt and rearranging, we get:

T = 2π / ω = 2π / (0.6435 radians)

Using a calculator, we find that T ≈ 9.759 seconds.

Therefore, the time at which the pendulum will be located at an angle of displacement of 8 degrees is approximately 9.759 seconds.

b) When will the pendulum return to its starting position?

Since the pendulum starts from rest at a maximum displacement of 10 degrees, and the pendulum will return to its starting position after completing one full oscillation, we can use the formula for the period of a simple pendulum:

T = 2π√(L/g)

Substituting the given values:

T = 2π√(0.350 m / 9.8 m/s^2)

Using a calculator, we find that T ≈ 2.239 seconds.

Therefore, the pendulum will return to its starting position after approximately 2.239 seconds.