How much work must be done to accelerate a baton from rest to an angular speed of 8.0 rad/s about its center. Consider the baton to be a uniform rod of length 0.84 m and mass 0.64 kg.
To find the work done to accelerate the baton, we can use the formula for rotational kinetic energy:
KE = (1/2) I ω^2
Where:
KE = Rotational Kinetic Energy
I = Moment of inertia
ω = Angular speed
First, let's calculate the moment of inertia of the baton. Since the baton is a uniform rod rotating about its center, the formula for the moment of inertia is:
I = (1/12) m L^2
Where:
m = Mass of the rod
L = Length of the rod
Plugging in the values:
m = 0.64 kg
L = 0.84 m
I = (1/12) * 0.64 kg * (0.84 m)^2
I ≈ 0.047 kg·m^2
Next, let's find the rotational kinetic energy using the given angular speed:
ω = 8.0 rad/s
KE = (1/2) * (0.047 kg·m^2) * (8.0 rad/s)^2
KE ≈ 1.216 J
Therefore, the work done to accelerate the baton from rest to an angular speed of 8.0 rad/s is approximately 1.216 Joules.
To calculate the work done to accelerate the baton, we need to know the moment of inertia of the rod. The moment of inertia of a uniform rod rotating about its center is given by the formula:
I = (1/12) * mL^2
Where:
I = moment of inertia
m = mass of the rod
L = length of the rod
Given:
m = 0.64 kg
L = 0.84 m
We can substitute these values into the formula to find the moment of inertia:
I = (1/12) * (0.64 kg) * (0.84 m)^2
I = 0.0244096 kg*m^2
The work done to accelerate the baton can be calculated using the formula:
Work = (1/2) * I * ω^2
Where:
Work = work done
I = moment of inertia
ω = angular speed
Given:
ω = 8.0 rad/s
Substituting the values into the formula, we get:
Work = (1/2) * (0.0244096 kg*m^2) * (8.0 rad/s)^2
Work = 0.781184 J
Therefore, the work done to accelerate the baton from rest to an angular speed of 8.0 rad/s about its center is approximately 0.781184 Joules.