A 12 g CD with a radius of 6.0 cm rotates with an angular speed of 40 rad/s. What is its kinetic energy?
What angular speed must the CD have if its kinetic energy is to be doubled?
To calculate the kinetic energy of the rotating CD, we can use the formula:
Kinetic Energy = (1/2) * I * ω^2
Where:
- Kinetic Energy represents the energy of an object due to its motion.
- I is the moment of inertia of the CD.
- ω (pronounced omega) is the angular speed of the CD in radians per second.
1) Calculating the kinetic energy:
First, we need to find the moment of inertia (I) of the CD.
The moment of inertia of a solid disk rotating about its center is given by the formula:
I = (1/2) * m * r^2
Where:
- m is the mass of the CD.
- r is the radius of the CD.
Given:
- The mass of the CD (m) is 12 g (which is equivalent to 0.012 kg).
- The radius of the CD (r) is 6.0 cm (which is equivalent to 0.06 m).
- The angular speed (ω) is 40 rad/s.
Let's calculate the moment of inertia (I):
I = (1/2) * m * r^2
= (1/2) * 0.012 kg * (0.06 m)^2
= 0.0000216 kg m^2
Now, let's calculate the kinetic energy using the formula mentioned earlier:
Kinetic Energy = (1/2) * I * ω^2
= (1/2) * 0.0000216 kg m^2 * (40 rad/s)^2
= 0.03456 J
Hence, the kinetic energy of the CD is 0.03456 Joules.
2) To find the angular speed required to double the kinetic energy:
Let's assume the new angular speed required to double the kinetic energy is ω_new.
We know that the kinetic energy is directly proportional to the square of the angular speed.
Thus, if we want to double the kinetic energy, we need to increase the angular speed by a factor of (√2).
So, the new angular speed (ω_new) can be calculated as:
ω_new = ω * √2
= 40 rad/s * √2
= 40 * 1.414
= 56.57 rad/s
Therefore, to double the kinetic energy, the CD must have an angular speed of approximately 56.57 rad/s.