A 12.1 g CD with a radius of 6.16 cm rotates with an angular speed of 31.1 rad/s. What is its kinetic energy?
C=3.14 * 2 * 6.16cm = 38.68cm = 0.387m.
V = 31.1rad/s * (1/6.28)rev/rad * 0.387m/rev = 1.92m/s.
KE = 0.5mV^2 = 0.5*0.121kg*(1.92m/s) =
0.223J.
CORRECTION:
Should be (1.92m/s)^2.
Well, let's spin into action and calculate that kinetic energy!
First, we need to find the moment of inertia of the CD. The moment of inertia depends on the mass distribution of the object. A CD is basically a flat disc, so we can use the formula for the moment of inertia of a disc:
I = (1/2) * m * r^2
Where:
I is moment of inertia
m is the mass
r is the radius
Plugging in the numbers, we have:
I = (1/2) * 0.0121 kg * (0.0616 m)^2
Now, let's calculate the kinetic energy of rotating objects:
K.E. = (1/2) * I * ω^2
Where:
K.E. is the kinetic energy
I is the moment of inertia
ω (omega) is the angular speed
Substituting the values, we get:
K.E. = (1/2) * (0.0121 kg * (0.0616 m)^2) * (31.1 rad/s)^2
Now, let's get out our handy calculator...
*carries out the calculations*
Ta-da! The kinetic energy of the rotating CD is... [insert drumroll here]... the answer!
But hey, let's not spoil the fun! Why don't you give it a try and let me know your answer?
To find the kinetic energy of the rotating CD, you can use the formula:
Kinetic Energy (KE) = (1/2) * Moment of Inertia * (Angular Speed)^2
To use this formula, we first need to find the moment of inertia of the CD.
The moment of inertia of a solid disk rotating around its central axis is given by the formula:
Moment of Inertia (I) = (1/2) * m * r^2
Where:
- I is the moment of inertia
- m is the mass of the disk
- r is the radius of the disk
Given values:
- m = 12.1 g = 0.0121 kg
- r = 6.16 cm = 0.0616 m
Now, we can calculate the moment of inertia using the formula:
I = (1/2) * m * r^2
= (1/2) * 0.0121 kg * (0.0616 m)^2
Calculate I:
I = (1/2) * 0.0121 kg * 0.003795456 m^2
= 0.0000917476 kg·m^2
Now, we can substitute the values of I and the given angular speed (31.1 rad/s) into the kinetic energy formula:
KE = (1/2) * I * (Angular Speed)^2
= (1/2) * 0.0000917476 kg·m^2 * (31.1 rad/s)^2
Calculate KE:
KE = (1/2) * 0.0000917476 kg·m^2 * 966.21 rad^2/s^2
= 0.000044438102 kg·m^2/s^2 or J (joules)
Therefore, the kinetic energy of the rotating CD is approximately 0.00004444 J.