A yo-yo is spun from rest by pulling on the string with a constant tension of 2.0 N. The radius of the inner rod on which the string is strung round is 0.50 cm. The tension is applied for 5.0 seconds after which the yo-yo is observed to spin with an angular velocity of 15 rad/sec.
a)What is the moment of inertia of the yo-yo?
b)What is the total angle the yo-yo has traveled through in these 5 seconds?
For a) I know the moment of inertia of the yo-yo (hollow cylinder) is 1/2*m*(r1^2+r2^2). We are given the inner radius, but I have no idea how to find the outer radius. I know how to find the mass since we know the tension and we can work backwards.
To find the outer radius of the yo-yo, we can use the given information about the tension and the applied force.
The tension in the string, 2.0 N, is equal to the force applied to spin the yo-yo. This force can be calculated using Newton's second law, which states that force is equal to mass multiplied by acceleration:
Tension = Mass * Acceleration
Since the yo-yo starts from rest and reaches an angular velocity of 15 rad/sec after 5.0 seconds, we know that the acceleration is equal to the change in angular velocity divided by the change in time:
Acceleration = Δω / Δt
Substituting in the given values, we have:
2.0 N = Mass * (15 rad/sec - 0 rad/sec) / 5.0 s
Simplifying, we can isolate the mass:
2.0 N = Mass * 3 rad/sec
Mass = 2.0 N / 3 rad/sec
Now that we have the mass, we can find the moment of inertia using the formula you mentioned:
Moment of Inertia = 1/2 * Mass * (r1^2 + r2^2)
Substituting in the given values for the inner radius (r1 = 0.50 cm), and assuming the outer radius is r2:
Moment of Inertia = 1/2 * (Mass) * (0.50 cm)^2 + r2^2
Now, to solve for the outer radius (r2), we can rearrange the formula:
(r2^2) = (Moment of Inertia - 1/2 * (Mass) * (0.50 cm)^2) / (Mass)
Solving this equation will give us the value of the outer radius (r2), and you can then use it to calculate the moment of inertia (making sure to convert units if necessary).
For part b), to find the total angle the yo-yo has traveled through in these 5 seconds, we can use the formula for angular displacement:
Angular Displacement = Angular Velocity * Time
Given that the angular velocity is 15 rad/sec and the time is 5.0 seconds, we can calculate the total angle.