A constant torque of 27.2 N · m is applied to

a grindstone for which the moment of inertia
is 0.158 kg · m2.
Find the angular speed after the grindstone
has made 18 rev.
Assume the grindstone
starts from rest.
Answer in units of rad/s

a = 27.2/.158 = 172 radians/second^2

18*2pi= 0 + 0t + (1/2)(172)t^2
113 = 86 t^2
t = 1.15 seconds

w = 0 + a t
w = 172*1.15 = 197 radians/second

To find the angular speed of the grindstone after it has made 18 revolutions, we can use the equation that relates torque, moment of inertia, and angular acceleration:

τ = Iα

Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

Since we're given the torque and the moment of inertia, we can rearrange the equation to solve for α:

α = τ / I

Plugging in the values:

α = 27.2 N · m / 0.158 kg · m^2

Now, we can use the equation that relates angular speed, angular acceleration, and time:

ω = α * t

Where ω is the angular speed, α is the angular acceleration, and t is the time.

Since the grindstone starts from rest, let's assume that it takes 18 revolutions to reach the desired angular speed. To convert 18 revolutions to time, we need to know the time it takes for one revolution. Let's assume that each revolution takes 1 second.

So, the total time taken for 18 revolutions is 18 * 1 second = 18 seconds.

Now, we can calculate the angular speed:

ω = α * t
ω = (27.2 N · m / 0.158 kg · m^2) * (18 seconds)

Calculating this, we get:

ω = 303.797 radians/second

Therefore, the angular speed of the grindstone after it has made 18 revolutions is 303.797 rad/s.

To find the angular speed of the grindstone, we can use the equation:

τ = I * α

where τ is the torque applied, I is the moment of inertia, and α is the angular acceleration.

In this case, the torque applied is 27.2 N · m, and the moment of inertia is 0.158 kg · m2. The angular acceleration, α, can be found using the equation:

α = Δω / Δt

where Δω is the change in angular speed and Δt is the time taken.

We are given that the grindstone starts from rest, so the initial angular speed, ω₀, is 0. We need to find the final angular speed, ω.

Now, we know that 1 revolution is equal to 2π radians. Therefore, 18 revolutions would be equal to:

18 * 2π = 36π radians

So, the change in angular speed, Δω, is ω - ω₀ = ω.

Using the equation Δω = α * Δt, we can rewrite it as ω = α * t.

To find the time taken, t, we can use the formula:

θ = ω₀ * t + 0.5 * α * t²

where θ is the angle covered, ω₀ is the initial angular speed, t is the time taken, and α is the angular acceleration.

In this case, the angle covered is 36π radians, the initial angular speed is 0, and α is unknown.

Since the grindstone starts from rest, the term ω₀ * t becomes 0, leading to:

36π = 0.5 * α * t²

Simplifying further:

72π = α * t²

Now, we can solve for α:

α = (72π) / t²

Substituting this value of α into the equation ω = α * t, we get:

ω = [(72π) / t²] * t

ω = 72π / t

Now we can solve for t using the equation:

θ = ω₀ * t + 0.5 * α * t²

Substituting the values θ = 36π and ω₀ = 0, we get:

36π = 0.5 * [(72π) / t²] * t²

Simplifying further:

36π = 36π

This shows that the value of t does not influence the equation, which means that any value of t will satisfy the equation.

Therefore, we can choose any value for t. Let's assume t = 1 second for simplicity.

Now, substituting t = 1 into the equation ω = 72π / t, we get:

ω = 72π / 1

ω = 72π rad/s

Hence, the angular speed of the grindstone after it has made 18 revolutions is 72π rad/s.