Motion of the Coin

The coin is rolling on a horizontal surface without sliding. This means that both translational motion (linear motion) and rotational motion are occurring simultaneously.

Given Information

- Mass of the coin: 4.8 kg
- Radius of the coin: 1 meter
- Angular velocity of the coin: 600 rotational min

Calculating Linear Velocity

To find the linear velocity of the coin, we can use the formula:

v = ω * r

where v is the linear velocity, ω (omega) is the angular velocity, and r is the radius of the coin.

In this case, the angular velocity is given in rotational min, so we need to convert it to radians per second.

1 rotational min = 2π radians

So, the angular velocity in radians per second is:

ω = (600 rotational min) * (2π radians / 1 rotational min) = 1200π radians per second

Now, substituting the values into the formula, we get:

v = (1200π radians per second) * (1 meter) = 1200π meters per second

Calculating Kinetic Energy

The total kinetic energy of the coin is the sum of its translational kinetic energy and rotational kinetic energy.

1. Translational Kinetic Energy:

The translational kinetic energy of an object can be calculated using the formula:

K_trans = (1/2) * m * v^2

where K_trans is the translational kinetic energy, m is the mass of the coin, and v is the linear velocity.

Substituting the values, we get:

K_trans = (1/2) * (4.8 kg) * (1200π meters per second)^2

2. Rotational Kinetic Energy:

The rotational kinetic energy of an object can be calculated using the formula:

K_rot = (1/2) * I * ω^2

where K_rot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

The moment of inertia for a solid sphere rolling without slipping is:

I = (2/5) * m * r^2

Substituting the values, we get:

I = (2/5) * (4.8 kg) * (1 meter)^2

Now, substituting the values into the formula, we get:

K_rot = (1/2) * [(2/5) * (4.8 kg) * (1 meter)^2] * [(1200π radians per second)^2]

Calculating the Total Kinetic Energy

To find the total kinetic energy, we simply add the translational and rotational kinetic energies together:

K_total = K_trans + K_rot

Finally, substituting the calculated values into the equation, we can find the total kinetic energy of the coin.