An electric motor exerts a constant torque of 10N⋅m to the shaft of a grindstone with mass 16kg and radius 0.5m. If the system starts from rest, find (a) the rotational kinetic energy of the grindstone after 8s, (b) the work done by the motor during this time, and (c) the average power delivered by the motor.
1. find moment if inertia I
2. angular acceleration a= torque/I
3. final angular speed= wf=a*time
4. rotalional KE= 1/2 I wf^2
5. angular displacement : wf^2=2 a*d or d= wf^2/2a or 1/2 wf^2/a
6. work done by motor=torque*d
7. average power=work done/time= torque*d/8
To solve this problem, we can use the following equations related to rotational motion:
(a) The rotational kinetic energy of an object is given by the equation:
K = (1/2) * I * ω²
where:
K is the rotational kinetic energy,
I is the moment of inertia of the object, and
ω is the angular velocity of the object.
(b) The work done on an object can be calculated by the equation:
W = τ * θ
where:
W is the work done,
τ is the torque applied to the object, and
θ is the angle through which the object has rotated.
(c) The average power delivered is given by:
P = W / t
where:
P is the average power delivered,
W is the work done, and
t is the time taken.
Now let's calculate the values step by step.
Given:
Torque (τ) = 10 N⋅m
Mass (m) = 16 kg
Radius (r) = 0.5 m
Time (t) = 8 s
(a) To find the rotational kinetic energy of the grindstone after 8 seconds, we need to calculate the angular velocity (ω) using the formula:
τ = I * α
where:
α is the angular acceleration.
Since the system starts from rest, the initial angular velocity (ω₀) is 0, and the final angular velocity (ω) can be calculated using the equation:
ω = ω₀ + α * t
= α * t
Since the torque (τ) is constant and equal to 10 N⋅m, we can use the equation:
τ = I * α
10 = I * α
We can calculate the moment of inertia (I) of the grindstone as:
I = (1/2) * m * r²
Substituting the given values:
I = (1/2) * 16 kg * (0.5 m)²
= 2 kg⋅m²
Now, we can substitute the value of I in the equation τ = 10 = I * α:
10 = 2 kg⋅m² * α
α = 5 rad/s²
Finally, substitute the value of α in the equation ω = α * t:
ω = (5 rad/s²) * (8 s)
= 40 rad/s
Now, we can calculate the rotational kinetic energy using the equation:
K = (1/2) * I * ω²
= (1/2) * 2 kg⋅m² * (40 rad/s)²
= 1 kg⋅m² * 1600 rad²/s²
= 1600 J
Therefore, the rotational kinetic energy of the grindstone after 8 seconds is 1600 Joules.
(b) To find the work done by the motor during this time, we need to calculate the angle (θ) through which the grindstone has rotated.
θ = ω₀ * t + (1/2) * α * t²
= 0 * 8 s + (1/2) * (5 rad/s²) * (8 s)²
= (1/2) * 5 rad/s² * 64 s²
= 160 rad
Now, we can calculate the work done using the equation:
W = τ * θ
= 10 N⋅m * 160 rad
= 1600 N⋅m
Therefore, the work done by the motor during this time is 1600 N⋅m.
(c) To find the average power delivered by the motor, we can use the equation:
P = W / t
= 1600 N⋅m / 8 s
= 200 W
Therefore, the average power delivered by the motor is 200 Watts.
To find the answers to these questions, we can use several key physics formulas. Let's break down the problem step by step:
(a) To find the rotational kinetic energy of the grindstone after 8 seconds, we need to calculate the angular acceleration and then use it to find the final angular velocity, which can be used to calculate the rotational kinetic energy.
First, let's find the angular acceleration. The formula for torque is given by:
Torque (τ) = I * α
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
In this case, the torque is 10 N·m and the moment of inertia can be calculated using the formula for a solid cylinder:
I = (1/2) * m * r^2
where m is the mass of the grindstone and r is its radius.
Plugging in the values: m = 16 kg and r = 0.5 m, we get:
I = (1/2) * 16 kg * (0.5 m)^2 = 2 kg·m^2
Now, rearranging the equation for torque, we can solve for α:
α = τ / I = 10 N·m / 2 kg·m^2 = 5 rad/s^2
Next, we can find the final angular velocity (ω) using the formula:
ω = ω0 + α * t
where ω0 is the initial angular velocity (which is zero in this case), α is the angular acceleration, and t is the time.
Plugging in the values: ω0 = 0 rad/s, α = 5 rad/s^2, and t = 8 s, we get:
ω = 0 + 5 rad/s^2 * 8 s = 40 rad/s
Finally, we can calculate the rotational kinetic energy (KE) using the formula:
KE = (1/2) * I * ω^2
Plugging in the values: I = 2 kg·m^2 and ω = 40 rad/s, we get:
KE = (1/2) * 2 kg·m^2 * (40 rad/s)^2 = 1600 J
So, the rotational kinetic energy of the grindstone after 8 seconds is 1600 Joules.
(b) To find the work done by the motor during this time, we can use the formula:
Work (W) = τ * θ
where W is the work done, τ is the torque, and θ is the angle through which the torque is applied.
In this case, the angle through which the torque is applied is equal to the angular displacement of the grindstone. The formula for angular displacement (θ) is:
θ = ω0 * t + (1/2) * α * t^2
where ω0 is the initial angular velocity (zero in this case), α is the angular acceleration, and t is the time.
Plugging in the values: ω0 = 0 rad/s, α = 5 rad/s^2, and t = 8 s, we get:
θ = 0 * 8 s + (1/2) * (5 rad/s^2) * (8 s)^2 = 160 rad
Now, we can calculate the work done:
W = τ * θ = 10 N·m * 160 rad = 1600 J
So, the work done by the motor during this time is 1600 Joules.
(c) To find the average power delivered by the motor, we can use the formula:
Average Power (P) = Work (W) / Time (t)
Plugging in the values: W = 1600 J and t = 8 s, we get:
P = 1600 J / 8 s = 200 W
So, the average power delivered by the motor is 200 Watts.