A spinning disk of mass 5.0 kg and radius 0.25m is rotating about an axis through its center at 400 rev/min.
a.) What is the angular velocity of the disk in rad/s?
b.) What is the moment of inertia of the disk?
c.) If it accelerates at a constant rate to 800 rev/min in 8 sec, what is its angular acceleration?
d.) What constant torque is required to produce this acceleration?
e.) When it reaches that speed, what is its kinetic energy?
f.) When it reaches that speed, what is its angular momentum?
a.) To find the angular velocity of the spinning disk in rad/s, we need to convert the given value of revolutions per minute (rev/min) to radians per second (rad/s). The conversion formula is:
Angular velocity (in rad/s) = Angular velocity (in rev/min) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Given:
Angular velocity = 400 rev/min
Substituting the value into the formula:
Angular velocity = 400 rev/min * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Simplifying the expression:
Angular velocity = (400 * 2π) / 60 rad/s
Calculating the value:
Angular velocity ≈ 41.9 rad/s
Therefore, the angular velocity of the spinning disk is approximately 41.9 rad/s.
b.) The moment of inertia of the disk is a measure of its resistance to changes in its rotational motion. For a solid disk rotating about its axis, the moment of inertia is given by the formula:
Moment of inertia (I) = (1/2) * mass * radius^2
Given:
Mass (m) = 5.0 kg
Radius (r) = 0.25 m
Substituting the values into the formula:
I = (1/2) * 5.0 kg * (0.25 m)^2
Calculating the value:
I = 0.625 kg·m^2
Therefore, the moment of inertia of the disk is 0.625 kg·m^2.
c.) To find the angular acceleration of the disk, we need to determine how its angular velocity changes over time. The formula for angular acceleration is:
Angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity = 400 rev/min
Final angular velocity = 800 rev/min
Time (t) = 8 sec
Converting the angular velocities to rad/s:
Initial angular velocity = 400 rev/min * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Final angular velocity = 800 rev/min * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Substituting the values into the formula:
α = (800 * (2π/60) - 400 * (2π/60)) / 8
Simplifying the expression:
α = (π/3) rad/s^2
Therefore, the angular acceleration of the spinning disk is π/3 rad/s^2.
d.) To find the constant torque required to produce the given angular acceleration, we use the formula:
Torque (τ) = moment of inertia * angular acceleration
Given:
Moment of inertia = 0.625 kg·m^2 (from part b)
Angular acceleration = π/3 rad/s^2 (from part c)
Substituting the values into the formula:
τ = 0.625 kg·m^2 * (π/3) rad/s^2
Calculating the value:
τ ≈ 0.654 N·m
Therefore, the constant torque required to produce the given angular acceleration is approximately 0.654 N·m.
e.) The kinetic energy of a rotating object is given by the formula:
Kinetic energy (KE) = (1/2) * moment of inertia * angular velocity^2
Given:
Moment of inertia = 0.625 kg·m^2 (from part b)
Angular velocity = 800 rev/min * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Converting the angular velocity to rad/s and substituting the values into the formula:
KE = (1/2) * 0.625 kg·m^2 * [(800 * (2π/60))^2]
Calculating the value:
KE ≈ 67,020 J
Therefore, when the spinning disk reaches the given speed, its kinetic energy is approximately 67,020 Joules.
f.) The angular momentum of a rotating object is given by the formula:
Angular momentum (L) = moment of inertia * angular velocity
Given:
Moment of inertia = 0.625 kg·m^2 (from part b)
Angular velocity = 800 rev/min * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Converting the angular velocity to rad/s and substituting the values into the formula:
L = 0.625 kg·m^2 * (800 * (2π/60))
Calculating the value:
L ≈ 209.44 kg·m^2/s
Therefore, when the spinning disk reaches the given speed, its angular momentum is approximately 209.44 kg·m^2/s.