A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 270 N applied to its edge causes the wheel to have an angular acceleration of 0.854 rad/s2.
(a) What is the moment of inertia of the wheel?
kg · m2
(b) What is the mass of the wheel?
kg
(c) If the wheel starts from rest, what is its angular velocity after 5.50 s have elapsed, assuming the force is acting during that time?
rad/s
a) τ = Iα
solve for I
b) Have to look up eq for moment of I for a solid cylinder. You have r and I
c) ω = αt
To find the moment of inertia of the wheel, we can use the formula:
Moment of inertia (I) = (mass (m) * radius squared (r^2))
Since we are given the radius of the wheel (r = 0.330 m), we need to find the mass of the wheel (m) first.
To find the mass of the wheel, we can use Newton's second law of motion, which states:
Force (F) = mass (m) * acceleration (a)
Rearranging the equation, we have:
m = F / a
Given the force (F = 270 N) and the angular acceleration (a = 0.854 rad/s^2), we can substitute the values to find the mass of the wheel.
(a) Calculating the moment of inertia (I):
I = m * r^2
(b) Calculating the mass of the wheel (m):
m = F / a
(c) Calculating the angular velocity (ω) after 5.50 s:
ω = ω₀ + (a * t)
where ω₀ is the initial angular velocity (which is 0 as the wheel starts from rest), a is the angular acceleration, and t is the time elapsed.
Substituting the values and solving for ω, we can find the angular velocity after 5.50 s have elapsed.
Let's calculate now:
(a) Moment of inertia (I):
I = m * r^2
(b) Mass of the wheel (m):
m = F / a
(c) Angular velocity (ω) after 5.50 s:
ω = ω₀ + (a * t)
To obtain the moment of inertia of the wheel, we can use the equation:
Angular acceleration (α) = Torque (τ) / Moment of Inertia (I)
Rearranging the equation, we have:
Moment of Inertia (I) = Torque (τ) / Angular acceleration (α)
In this case, the torque is given by multiplying the applied force by the radius of the wheel:
Torque (τ) = Force (F) x Radius (r)
Substituting the known values, we have:
τ = 270 N x 0.330 m
To find the moment of inertia (I), we divide the torque by the angular acceleration:
I = (270 N x 0.330 m) / 0.854 rad/s²
Calculating this gives us the value of the moment of inertia in kg · m².
To find the mass of the wheel, we can use the equation:
Moment of Inertia (I) = Mass (m) x Radius (r)²
Rearranging the equation, we can solve for the mass:
Mass (m) = Moment of Inertia (I) / Radius (r)²
Substituting the known values, we have:
m = I / (r²)
Calculating this gives us the mass of the wheel in kg.
To calculate the angular velocity after 5.50 seconds, we can use the equation:
Final angular velocity (ω) = Initial angular velocity (ω₀) + Angular acceleration (α) x Time (t)
Since the wheel starts from rest, the initial angular velocity is zero:
ω₀ = 0 rad/s
Substituting the known values, we have:
ω = 0 + (0.854 rad/s²) x (5.50 s)
Calculating this gives us the angular velocity of the wheel in rad/s.