A solid cylinder of mass 20 kg rotates about its axis with angular speed of 100 rad/sec. the radius of the cylinder is 0.25m. what is the kinetic energy of the cylinder?
Why did the solid cylinder start rotating? It wanted to win the "Spinny-est Object of the Year" award! Let's calculate its kinetic energy.
The formula for the kinetic energy of a rotating object is given by:
KE = (1/2) I ω^2,
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular speed.
For a solid cylinder rotating about its axis, the moment of inertia is given by:
I = (1/2) m r^2,
where m is the mass of the cylinder and r is its radius.
Plugging in the given values:
m = 20 kg,
r = 0.25 m,
ω = 100 rad/sec,
Let's crunch the numbers:
I = (1/2) x 20 kg x (0.25 m)^2,
Simplifying:
I = 0.625 kg m^2.
Now, let's calculate the kinetic energy:
KE = (1/2) x 0.625 kg m^2 x (100 rad/sec)^2.
Calculating further:
KE = 3125 J.
Therefore, the kinetic energy of the cylinder is 3125 Joules. Happy spinning!
To calculate the kinetic energy of the cylinder, you can use the formula:
Kinetic Energy = 1/2 * moment of inertia * angular velocity^2
1. Calculate the moment of inertia (I) of the cylinder. For a solid cylinder rotating about its axis, the moment of inertia is given by:
I = 1/2 * m * r^2
where:
m = mass of the cylinder
r = radius of the cylinder
In this case, the mass of the cylinder (m) is 20 kg, and the radius (r) is 0.25 m.
I = 1/2 * 20 kg * (0.25 m)^2
2. Substitute the values into the equation to find the moment of inertia (I):
I = 1/2 * 20 kg * (0.0625 m^2)
= 0.625 kg m^2
3. Calculate the kinetic energy using the moment of inertia and angular velocity:
Kinetic Energy = 1/2 * I * (angular velocity)^2
Substituting the values we have:
Kinetic Energy = 1/2 * 0.625 kg m^2 * (100 rad/sec)^2
Calculating further:
Kinetic Energy = 1/2 * 0.625 kg m^2 * 10,000 rad^2/sec^2
= 3125 J
Therefore, the kinetic energy of the cylinder is 3125 Joules.
To find the kinetic energy of the cylinder, we can use the formula:
Kinetic Energy = 1/2 * I * ω^2
Where:
- Kinetic Energy is the energy possessed by the rotating object due to its motion.
- I is the moment of inertia of the object.
- ω is the angular speed of the object.
To find the moment of inertia of a solid cylinder, we use the formula:
I = (1/2) * m * r^2
Where:
- m is the mass of the cylinder.
- r is the radius of the cylinder.
Let's plug in the given values:
m = 20 kg
r = 0.25 m
ω = 100 rad/sec
First, calculate the moment of inertia:
I = (1/2) * 20 kg * (0.25 m)^2
I = 0.125 kg * (0.0625 m^2)
I = 0.0078125 kg * m^2
Next, substitute the moment of inertia and angular speed into the kinetic energy formula:
Kinetic Energy = 1/2 * 0.0078125 kg * m^2 * (100 rad/sec)^2
Kinetic Energy = 1/2 * 0.0078125 kg * m^2 * 10,000 rad^2/sec^2
Kinetic Energy = 0.390625 kg * m^2 * rad^2/sec^2
Hence, the kinetic energy of the cylinder is approximately 0.391 J (joules).