Parth G | You (Probably) Haven't Been Told About the Second Type of Kinetic Energy @ParthGChannel | Uploaded 2 years ago | Updated 11 hours ago
How can we calculate the rotational kinetic energy of any object? We will look at how this kinetic energy relates to angular speed (or angular velocity) and the moment of inertia of the object.
On this channel, I've not properly discussed rotational kinetic energy. In addition to this, I've often referred to linear kinetic energy as simply "kinetic energy". So today we're setting the record straight.
Any object with angular motion (e.g. moving along a curved path or spinning about an axis) will have angular kinetic energy. This energy depends on the speed at which the object spins, the mass of the object, and the shape of the object too.
The angular speed of our object (i.e. magnitude of angular velocity) simply measures the angle the rotation covers in a unit of time. For example, our object could be moving at 45 degrees per second, or pi/4 radians per second. Generally we prefer to use radians as our angular unit.
The moment of inertia of our object accounts for both the mass and the shape of the object. It can be calculated by taking a very small chunk of mass making up the object, and multiplying this by the square of the perpendicular distance between the chunk of mass and the axis of rotation. Then we do this for all chunks of mass making up the object, and add up all these contributions.
In other words, the moment of inertia of an object can be found by calculating the integral of the square of the distance between the mass and the rotation axis, with respect to the mass of the object. Interestingly this tells us that two objects may have the same external shape, but if one is hollow and the other is solid / filled, they will have different moments of inertia. Also, an object will have a different moment of inertia depending on what axis we intend to spin it about!
We can therefore think of the moment of inertia as a measure of our object's resistance to angular motion. Or it's a measure of how much torque is needed to have our object experience a given angular acceleration. This is similar to how an object's mass is a measure of its resistance to linear motion, or how much force is needed for a specific linear acceleration.
In general there are many similarities between moment of inertia (angular) and mass (linear), or angular speed and linear speed, or other angular and linear quantities. For our solid spherical ball, the moment of inertia is given by (2/5)MR^2 where M is the mass of the sphere and R is its radius.
And the rotational kinetic energy of any object is given by finding (1/2)Iw^2 where I is the moment of inertia, and w (or omega) is the angular speed. This equation is similar to the linear kinetic energy equation, (1/2)mv^2. Both quantities are energies, so are measured in joules.
For our specific foam ball, we find its mass and radius by measuring these the usual way, and the angular speed by putting a dot on the ball and timing how long it takes to complete one full oscillation. Then we can combine all this information to work out how much rotational kinetic energy it has as a result of spinning. We also work out the speed the ball would move with if it had an equivalent amount of linear kinetic energy!
Thanks so much for watching - please do check out my socials here:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Music Chanel - Parth G's Shenanigans
Merch - https://parth-gs-merch-stand.creator-spring.com/
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - Rotational Kinetic Energy
1:44 - The Equation, Moment of Inertia, and Angular Speed / Velocity
2:04 - Understanding Angular Velocity
3:19 - Moment of Inertia - How Hard is it to Spin a Ball?
How can we calculate the rotational kinetic energy of any object? We will look at how this kinetic energy relates to angular speed (or angular velocity) and the moment of inertia of the object.
On this channel, I've not properly discussed rotational kinetic energy. In addition to this, I've often referred to linear kinetic energy as simply "kinetic energy". So today we're setting the record straight.
Any object with angular motion (e.g. moving along a curved path or spinning about an axis) will have angular kinetic energy. This energy depends on the speed at which the object spins, the mass of the object, and the shape of the object too.
The angular speed of our object (i.e. magnitude of angular velocity) simply measures the angle the rotation covers in a unit of time. For example, our object could be moving at 45 degrees per second, or pi/4 radians per second. Generally we prefer to use radians as our angular unit.
The moment of inertia of our object accounts for both the mass and the shape of the object. It can be calculated by taking a very small chunk of mass making up the object, and multiplying this by the square of the perpendicular distance between the chunk of mass and the axis of rotation. Then we do this for all chunks of mass making up the object, and add up all these contributions.
In other words, the moment of inertia of an object can be found by calculating the integral of the square of the distance between the mass and the rotation axis, with respect to the mass of the object. Interestingly this tells us that two objects may have the same external shape, but if one is hollow and the other is solid / filled, they will have different moments of inertia. Also, an object will have a different moment of inertia depending on what axis we intend to spin it about!
We can therefore think of the moment of inertia as a measure of our object's resistance to angular motion. Or it's a measure of how much torque is needed to have our object experience a given angular acceleration. This is similar to how an object's mass is a measure of its resistance to linear motion, or how much force is needed for a specific linear acceleration.
In general there are many similarities between moment of inertia (angular) and mass (linear), or angular speed and linear speed, or other angular and linear quantities. For our solid spherical ball, the moment of inertia is given by (2/5)MR^2 where M is the mass of the sphere and R is its radius.
And the rotational kinetic energy of any object is given by finding (1/2)Iw^2 where I is the moment of inertia, and w (or omega) is the angular speed. This equation is similar to the linear kinetic energy equation, (1/2)mv^2. Both quantities are energies, so are measured in joules.
For our specific foam ball, we find its mass and radius by measuring these the usual way, and the angular speed by putting a dot on the ball and timing how long it takes to complete one full oscillation. Then we can combine all this information to work out how much rotational kinetic energy it has as a result of spinning. We also work out the speed the ball would move with if it had an equivalent amount of linear kinetic energy!
Thanks so much for watching - please do check out my socials here:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Music Chanel - Parth G's Shenanigans
Merch - https://parth-gs-merch-stand.creator-spring.com/
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - Rotational Kinetic Energy
1:44 - The Equation, Moment of Inertia, and Angular Speed / Velocity
2:04 - Understanding Angular Velocity
3:19 - Moment of Inertia - How Hard is it to Spin a Ball?
