Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm.

Q: A motor spins up the flywheel with a constant torque of 50 {\rm N \cdot m}. How long does it take the flywheel to reach top speed? In units of seconds.

Torque = (moment of inertia) x (angular acceleration) = 50 N*m

The moment of inertia is
I = (1/2) M*R^2 = 281.25 kg*m^2

The maximum angular velocity you want to achieve is
w = 1200 rev/min * (2 pi/60) = 125.66 rad/s

The angular acceleration rate is
alpha = 125.66 rad/s / t
where t is the time required.

alpha = 50/281.25 rad/s^2 = 125.66/t

Solve for t in seconds

just make sure to use r, not d. like drwls.

To solve this problem, we can use the formula for angular acceleration:

α = τ / I

Where:
α = angular acceleration
τ = torque
I = moment of inertia

We can calculate the moment of inertia of the flywheel using the formula:

I = (1/2) * m * r^2

Where:
m = mass of the flywheel
r = radius of the flywheel

Let's calculate the moment of inertia first:

m = 250 kg
r = (1.5 m) / 2 = 0.75 m

I = (1/2) * 250 kg * (0.75 m)^2 = 70.3125 kg * m^2

Now we can calculate the angular acceleration:

α = 50 N * m / 70.3125 kg * m^2 = 0.711109524 rad/s^2

To find the time it takes to reach top speed, we need to know the angular velocity at the start. In this case, we assume it starts from rest, so ω_0 = 0 rad/s.

We can use the equation for angular velocity:

ω = ω_0 + α * t

Where:
ω = final angular velocity
ω_0 = initial angular velocity
t = time

Since ω_0 = 0, the equation simplifies to:

ω = α * t

We want to find the time it takes (t) when the final angular velocity (ω) is 1200 rpm. First, we need to convert 1200 rpm to rad/s:

ω = 1200 rpm * (2π rad/1 min) * (1 min/60 s) = 125.6637 rad/s

Now we can rearrange the equation to solve for time (t):

t = ω / α = (125.6637 rad/s) / (0.711109524 rad/s^2) = 176.8082 s

Therefore, it takes approximately 176.8082 seconds for the flywheel to reach top speed.

To find the time it takes for the flywheel to reach top speed, we need to use the laws of rotational motion. The relevant equation is:

τ = Iα

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

First, let's calculate the moment of inertia of the flywheel using the formula:

I = ½ * m * r^2

where m is the mass of the flywheel and r is the radius of the flywheel.

Given:
Mass of the flywheel (m) = 250 kg
Diameter of the flywheel = 1.5 m

We need to convert the diameter to radius:
Radius (r) = diameter/2 = 1.5 m/2 = 0.75 m

Plugging the values into the formula, we get:
I = ½ * 250 kg * (0.75 m)^2 = 70.3125 kg·m^2

Now, let's calculate the angular acceleration (α) using the formula:

α = τ / I

Given:
Torque (τ) = 50 N·m

Plugging in the values, we get:
α = 50 N·m / 70.3125 kg·m^2 ≈ 0.71 rad/s^2

Now, we can use the formula for angular velocity (ω) to find the time (t) it takes for the flywheel to reach top speed:

ω = αt

Given:
Maximum angular velocity (ω) = 1200 rpm = 1200 revolutions per minute = 1200/60 revolutions per second = 20 revolutions per second

We need to convert revolutions per second to radians per second:
1 revolution = 2π radians
So, 20 revolutions/second = 20 * 2π radians/second ≈ 125.66 radians/second

Plugging in the values, we get:
125.66 radians/second = 0.71 rad/s^2 * t

Now, rearranging the equation to solve for time (t):
t = ω / α
= 125.66 radians/second / 0.71 rad/s^2 ≈ 177.46 seconds

Therefore, it takes approximately 177.46 seconds for the flywheel to reach top speed when spun up with a constant torque of 50 N·m.