Consider a 9·kg disk with a radius of 0.9·m and rotating about an axis passing through its center.
(a) What is its rotational inertia? ______ kg·m2 ?
(b) Suppose the disk is rotating counterclockwise and speeding up with a rotational acceleration of 4·rad/s2. What is the net torque acting on the disk? _____N·m?
what direction clockwise or counterclockwise?
(c) Suppose the torque you found for the last part is being produced by a force applied at a point 0.8·m from the axis. If the direction of the force is perpendicular to a line from the axis to that point, what is the magnitude of the force? _____N?
To find the answers to these questions, we need to use the concepts of rotational inertia, torque, and Newton's second law of rotational motion.
(a) The rotational inertia of a disk can be calculated using the formula:
Inertia = (1/2) * Mass * Radius^2
Given:
Mass (m) = 9 kg
Radius (r) = 0.9 m
Using the formula, we can calculate the rotational inertia:
Inertia = (1/2) * 9 kg * (0.9 m)^2
Inertia = 3.645 kg·m^2
Therefore, the rotational inertia of the disk is 3.645 kg·m^2.
(b) To find the net torque acting on the disk, we can use the formula:
Torque (τ) = Inertia * Angular Acceleration
Given:
Inertia (I) = 3.645 kg·m^2
Angular Acceleration (α) = 4 rad/s^2
Using the formula, we can calculate the net torque:
Torque = 3.645 kg·m^2 * 4 rad/s^2
Torque = 14.58 N·m (counterclockwise)
The net torque acting on the disk is 14.58 N·m in the counterclockwise direction.
(c) To find the magnitude of the force producing the torque, we can rearrange the torque formula:
Torque (τ) = Force (F) * Distance (r)
Given:
Torque (τ) = 14.58 N·m
Distance (r) = 0.8 m
Using the formula and rearranging it, we can find the magnitude of the force:
Force = Torque / Distance
Force = 14.58 N·m / 0.8 m
Force = 18.225 N
Therefore, the magnitude of the force producing the torque is 18.225 N.
(a) The rotational inertia of a disk can be calculated using the formula:
I = (1/2) * m * r^2
where I is the rotational inertia, m is the mass of the disk, and r is the radius of the disk.
Given that the mass of the disk is 9 kg and the radius is 0.9 m, we can substitute these values into the formula:
I = (1/2) * 9 kg * (0.9 m)^2
I = 0.5 * 9 kg * 0.81 m^2
I = 3.645 kg·m^2
Therefore, the rotational inertia of the disk is 3.645 kg·m^2.
(b) The net torque acting on the disk can be calculated using the formula:
τ = I * α
where τ is the net torque, I is the rotational inertia, and α is the rotational acceleration.
Given that the rotational inertia is 3.645 kg·m^2 and the rotational acceleration is 4 rad/s^2, we can substitute these values into the formula:
τ = 3.645 kg·m^2 * 4 rad/s^2
τ = 14.58 N·m
The net torque acting on the disk is 14.58 N·m. Since the disk is rotating counterclockwise, the net torque is in the counterclockwise direction.
(c) To find the magnitude of the force producing the torque, we can rearrange the formula for torque to solve for force:
τ = r * F * sin(θ)
where τ is the torque, r is the distance from the axis to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the line from the axis to the point.
Since the torque is 14.58 N·m and the distance from the axis to the point is 0.8 m, we can substitute these values into the formula:
14.58 N·m = 0.8 m * F * sin(90°)
F * sin(90°) = 14.58 N·m / 0.8 m
F = (14.58 N·m / 0.8 m) / sin(90°)
F = 18.225 N
The magnitude of the force is 18.225 N.