A typical ten-pound car wheel has a moment of inertia of about 0.35 kg*m^2. The wheel rotates about the axle at a constant angular speed making 45.0 full revolutions in a time interval of 7.00 sec .
What is the rotational kinetic energy of the rotating wheel?
Angular velocity, ω
= (45*2π/7) radians / sec.
Rotational kinetic energy
= (1/2)Iω²
= (1/2)0.35(45*2π/7)² J.
= ?
To find the rotational kinetic energy of the rotating wheel, we can use the formula:
Rotational kinetic energy (K) = (1/2) * moment of inertia * (angular speed)^2
Given:
Moment of inertia (I) = 0.35 kg*m^2
Angular speed (ω) = (45.0 revolutions) / (7.00 sec)
First, we need to convert the angular speed from revolutions per second to radians per second:
1 revolution = 2π radians
So, (angular speed in radians) = (angular speed in revolutions) * (2π radians)
ω = (45.0 revolutions) * (2π radians) / (7.00 sec)
Now, we can substitute the values into the formula and calculate the rotational kinetic energy:
K = (1/2) * 0.35 kg*m^2 * [(45.0 revolutions) * (2π radians) / (7.00 sec)]^2
Let's calculate this:
To find the rotational kinetic energy of the rotating wheel, we can use the formula:
Rotational Kinetic Energy (KE) = (1/2) * moment of inertia * angular velocity^2
First, let's find the angular velocity.
The number of full revolutions, 45.0, can be converted into radians by multiplying it by 2π (since 1 revolution = 2π radians).
Number of radians = 45.0 * 2π = 90π radians
The time interval, 7.00 sec, gives us the time for 45.0 revolutions.
To find the angular velocity, divide the total angle in radians by the time interval:
Angular velocity (ω) = (Number of radians) / (Time interval)
ω = (90π radians) / (7.00 sec)
Now that we have the angular velocity, we can substitute it into the formula for rotational kinetic energy:
KE = (1/2) * moment of inertia * (angular velocity)^2
Substituting the given values:
KE = (1/2) * 0.35 kg*m^2 * [(90π radians) / (7.00 sec)]^2
Calculating this expression will give us the rotational kinetic energy of the rotating wheel.