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) Convert revolutions per minute to radians per second. Surely you can do that.
b) The moment of inertia of a disc of radius R and mass M is
I = (1/2)M*R^2
c) Convert 800 rev/min to
w = ___ radians/sec, and divide that by 8.0 seconds for the angular acceleration, ____rad/s^2.
d) Torque = (Moment of inertia)*(angular accleration)
e) (1/2) I w^2
f) I*w
You do the numbers.
To solve the given problems, we'll utilize the formulas and principles of rotational motion. Let's break down each question and explain how to find the answers:
a.) What is the angular velocity of the disk in rad/s?
Angular velocity is defined as the rate of change of the angle in radians with respect to time. To convert from revolutions per minute (rev/min) to radians per second (rad/s), we need to use the conversion factor of 2π rad = 1 revolution and 60 sec = 1 min.
So, to find the angular velocity, we can use the formula:
Angular velocity (ω) = (2π * N) / T,
where N is the number of revolutions and T is the time in seconds.
Given:
Number of revolutions (N) = 400 rev/min
Time in seconds (T) = 1 min = 60 sec
Substituting the values into the formula:
ω = (2π * 400) / 60 rad/s
Calculating the value:
ω = (800π) / 60 rad/s
Therefore, the angular velocity of the disk is (800π / 60) rad/s.
b.) What is the moment of inertia of the disk?
The moment of inertia (I) represents the rotational analog of mass in linear motion. It depends on the mass distribution and the shape of the object.
For a uniform disk rotating about an axis through its center, the moment of inertia can be calculated using the formula:
Moment of inertia (I) = (1/2) * m * r^2,
where m is the mass of the disk and r is the radius.
Given:
Mass of the disk (m) = 5.0 kg
Radius of the disk (r) = 0.25 m
Substituting the values into the formula:
I = (1/2) * 5.0 * 0.25^2 kg.m^2
Calculating the value:
I = (1/2) * 5.0 * 0.0625 kg.m^2
Therefore, the moment of inertia of the disk is 0.15625 kg.m^2.
c.) If it accelerates at a constant rate to 800 rev/min in 8 sec, what is its angular acceleration?
Angular acceleration (α) represents the rate of change of angular velocity with respect to time. It is given by the formula:
Angular acceleration (α) = (ωf - ωi) / t,
where ωf is the final angular velocity, ωi is the initial angular velocity and t is the time interval.
Given:
Initial angular velocity (ωi) = (800π / 60) rad/s
Final angular velocity (ωf) = 400 rev/min (converted to rad/s using the formula above)
Time interval (t) = 8 sec
Substituting the values into the formula:
α = ((400π / 60) - (800π / 60)) / 8 rad/s^2
Simplifying the expression:
α = (-400π / 60) / 8 rad/s^2
Calculating the value:
α = (-50π / 6) rad/s^2
Therefore, the angular acceleration of the disk is (-50π / 6) rad/s^2.
d.) What constant torque is required to produce this acceleration?
Torque (τ) is defined as the moment of force that causes an object to rotate about an axis. The torque required to produce an angular acceleration can be determined using the equation:
Torque (τ) = I * α,
where I is the moment of inertia and α is the angular acceleration.
Given:
Moment of inertia (I) = 0.15625 kg.m^2
Angular acceleration (α) = (-50π / 6) rad/s^2
Substituting the values into the formula:
τ = 0.15625 * (-50π / 6) N.m
Calculating the value:
τ = -2.604π N.m (approximately)
Therefore, the constant torque required to produce this acceleration is approximately -2.604π N.m.
e.) When it reaches that speed, what is its kinetic energy?
The kinetic energy (KE) of a rotating object is given by the formula:
Kinetic energy (KE) = (1/2) * I * ω^2,
where I is the moment of inertia and ω is the angular velocity.
Given:
Moment of inertia (I) = 0.15625 kg.m^2
Angular velocity (ω) = (800π / 60) rad/s (converted from 800 rev/min using the formula above)
Substituting the values into the formula:
KE = (1/2) * 0.15625 * ((800π / 60)^2) J
Calculating the value:
KE ≈ 2109.24 J (approximately)
Therefore, when it reaches that speed, the kinetic energy of the disk is approximately 2109.24 J.
f.) When it reaches that speed, what is its angular momentum?
Angular momentum (L) of a rotating body is given by the formula:
Angular momentum (L) = I * ω,
where I is the moment of inertia and ω is the angular velocity.
Given:
Moment of inertia (I) = 0.15625 kg.m^2
Angular velocity (ω) = (800π / 60) rad/s (converted from 800 rev/min using the formula above)
Substituting the values into the formula:
L = 0.15625 * (800π / 60) kg.m^2/s
Calculating the value:
L ≈ 260.416π kg.m^2/s (approximately)
Therefore, when it reaches that speed, the angular momentum of the disk is approximately 260.416π kg.m^2/s.