How much work must be done to accelerate a baton from rest to an angular speed of 7.27 rad/s about its center? Consider the baton to be a uniform rod of length 0.529 m and mass 0.478 kg

The moment of inertia of a uniform rod, about a line through the center and perpendicular to the rod, is

I = M L^2/12

The kinetic energy of the rotating rod is (I/2)* w^2

w = 7.27 rad/s L = 0.529 m
M = 0.478 kg

Calculate that kinetic energy. It equals the work required.

Well, I have to say, that baton must be really desperate to get itself spinning! So, let's calculate the amount of work required.

In order to find the work done to accelerate the baton, we need to use the formula for rotational kinetic energy. The formula is K = (1/2)Iω^2, where K is the kinetic energy, I is the moment of inertia, and ω is the angular speed.

To find the moment of inertia for a uniform rod rotating about its center, we use the formula I = (1/12)mL^2, where m is the mass of the baton and L is its length.

Plugging in the given values, we have:
I = (1/12)(0.478 kg)(0.529 m)^2
I ≈ 0.00771 kg·m^2

Now, let's calculate the work done:
K = (1/2)(0.00771 kg·m^2)(7.27 rad/s)^2
K ≈ 0.198 J

So, the baton needs approximately 0.198 Joules of work to reach an angular speed of 7.27 rad/s. That's quite a workout for a little baton! I hope it's been hitting the gym!

To find the work done to accelerate the baton, we need to determine the moment of inertia of the baton and then use the equation for rotational kinetic energy.

Step 1: Find the moment of inertia (I) of the baton:
The moment of inertia for a uniform rod rotating about its center is given by the equation:
I = (1/12) * m * L^2
where m is the mass of the rod and L is its length.

Given:
Mass of the baton (m) = 0.478 kg
Length of the baton (L) = 0.529 m

Substituting the values into the equation, we get:
I = (1/12) * 0.478 kg * (0.529 m)^2
I = 0.0102 kg·m^2

Step 2: Use the equation for rotational kinetic energy (K) to find the work (W) done:
The equation for rotational kinetic energy is:
K = (1/2) * I * w^2
where w is the angular speed of the baton.

We are given:
Angular speed of the baton (w) = 7.27 rad/s
Moment of inertia of the baton (I) = 0.0102 kg·m^2

Substituting the values into the equation, we get:
K = (1/2) * 0.0102 kg·m^2 * (7.27 rad/s)^2
K = 0.267 J

Since the work done to accelerate an object is equal to the change in its kinetic energy, the work done is equal to the rotational kinetic energy (K):
W = K = 0.267 J

Therefore, the work done to accelerate the baton from rest to an angular speed of 7.27 rad/s about its center is 0.267 J.

To determine the amount of work needed to accelerate the baton, we need to calculate the rotational kinetic energy (KE) of the baton.

The formula for rotational kinetic energy is:

KE = (1/2) * I * ω^2

Where:
- KE is the rotational kinetic energy
- I is the moment of inertia of the baton
- ω is the angular velocity of the baton

In this case, the baton is considered to be a uniform rod. The moment of inertia of a uniform rod rotating about its center is given by the formula:

I = (1/12) * m * L^2

Where:
- m is the mass of the baton
- L is the length of the baton

Now, let's calculate the moment of inertia (I):

I = (1/12) * (0.478 kg) * (0.529 m)^2
I = 0.00430 kg·m^2

Next, substitute the values of I and ω into the formula for KE:

KE = (1/2) * (0.00430 kg·m^2) * (7.27 rad/s)^2
KE = 0.114 J (joules)

The work done to accelerate the baton is equal to the change in kinetic energy:

Work = KE_final - KE_initial

Since the baton is initially at rest (ω_initial = 0), the initial kinetic energy (KE_initial) is zero. Therefore:

Work = KE_final - 0
Work = KE_final

So, the amount of work needed to accelerate the baton from rest to an angular speed of 7.27 rad/s about its center is 0.114 J (joules).