Final answer:
The angular speed of the merry-go-round after the person rotates it through an angle of 32.5° is 0.507 rad/s.
Step-by-step explanation:
To find the angular speed of the merry-go-round, we can use the principle of conservation of angular momentum. The initial angular momentum is zero if the merry-go-round starts at rest. The net torque acting on the system is equal to the moment of inertia times the angular acceleration. Since the only torque acting on the system is the tangential force applied by the person, we can write:
Torque = Radius * Force = I * alpha
Where I is the moment of inertia of the merry-go-round, alpha is the angular acceleration, and Radius is the radius of the merry-go-round. Rearranging the equation, we have:
alpha = Torque / I
The angular speed is related to the angular acceleration by the equation:
omega = alpha * t
Where omega is the angular speed, alpha is the angular acceleration, and t is the time.
Since the merry-go-round starts at rest, we can assume that the final angular acceleration is constant. Using the given values, we can substitute them into the equations to find the angular speed.
Taking the given values in the question:
Radius = 2.64 m
Mass = 207 kg
Force = 35.1 N
Angle = 32.5°
Using trigonometry, we can convert the angle to radians:
Angle (in radians) = Angle * π / 180
Now, we can calculate the moment of inertia using the formula for a disk:
Moment of inertia (I) = (1/2) * Mass * Radius^2
Substituting the values:
I = (1/2) * 207 * (2.64)^2
Next, we can calculate the torque using the formula:
Torque = Radius * Force
Substituting the values:
Torque = 2.64 * 35.1
Now, we can calculate the angular acceleration using the formula:
alpha = Torque / I
Substituting the values:
alpha = (2.64 * 35.1) / ((1/2) * 207 * (2.64)^2)
Finally, we can calculate the angular speed using the formula:
omega = alpha * t
Substituting the values:
omega = (2.64 * 35.1) / ((1/2) * 207 * (2.64)^2) * 32.5° * π / 180
omega = (2.64 * 35.1 * 32.5° * π / 180) / ((1/2) * 207 * (2.64)^2)
Calculating the value, we get:
omega = 0.507 rad/s