Indiana Jones is swinging from a rope. The distance between the pivot point and his center of mass is 31.00m. He begins swinging from rest at an angle theta=20.00 degrees. Assuming that Indiana and the rope can be treated as a simple pendulum, what is the value of theta after 1.390s (in degrees)?
First calculate the period of a complete swing back and forth. It is
P = 2 pi sqrt(L/g)
which is about ten seconds
In 1.39 seconds, he will not have reached the lowest position (zero degrees) yet.
Theta ( in degrees) = 20 cos (2 pi*t/P)
Use your P and t to calculate theta at that time
To determine the value of theta after 1.390s, we can apply the equations of motion for a simple pendulum.
The equation for the period of a simple pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
In this case, the length of the pendulum is the distance between the pivot point and Indiana's center of mass, which is 31.00m.
Next, we can find the value of g. The standard value for the acceleration due to gravity is approximately 9.8 m/s^2.
Now, we can substitute the values into the equation T = 2π√(L/g) to find the period of the pendulum.
T = 2π√(31.00/9.8) ≈ 11.178s
The period of the pendulum is approximately 11.178s.
To find the angle theta after 1.390s, we can use the equation θ = θ0 * cos((2π*t)/T), where θ is the angular displacement, θ0 is the initial angle, t is the time, and T is the period.
In this case, θ0 is given as 20.00 degrees and t is 1.390s.
θ = 20.00 * cos((2π*1.390)/11.178)
Calculating this expression, we find the value of theta after 1.390s is approximately 13.135 degrees (rounded to three decimal places).
To find the value of theta after 1.390 seconds, we can use the equation of motion for a simple pendulum:
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
In this case, the length of the pendulum is the distance between the pivot point and Indiana Jones' center of mass, which is given as 31.00m.
First, let's calculate the period of the pendulum. We know that the period is the time it takes for the pendulum to complete one full swing, which is the given time, 1.390s, divided by the number of swings.
T = 1.390s
Next, we need to calculate the acceleration due to gravity, g. This can be approximated as 9.8 m/s^2.
g = 9.8 m/s^2
Using the equation for the period, we can rearrange it to solve for the length of the pendulum:
L = (T/(2π))^2 * g
Now we can substitute the values and solve for L:
L = (1.390s/(2π))^2 * 9.8 m/s^2
L ≈ 11.6809 m
Now that we have the length of the pendulum, we can use the equation for the angular displacement in a simple pendulum:
θ = θ₀ * cos(√(g/L) * t)
Where:
θ is the angular displacement
θ₀ is the initial angular displacement
g is the acceleration due to gravity
L is the length of the pendulum
t is the time
In this case, the initial angular displacement is given as 20.00 degrees.
θ₀ = 20.00 degrees
Substituting the values into the equation and solving for θ:
θ = 20.00 degrees * cos(√(9.8 m/s^2 / 11.6809 m) * 1.390s)
θ ≈ 6.6963 degrees
Therefore, the value of theta after 1.390 seconds is approximately 6.6963 degrees.