a uniform disk of mass 3kg and radius 12cm rotates at 480 rev/min. find its kinetic energy (Hint : I of uniform disk =1/2MR2)
KE=1/2 I w^2
angular speed= PI*rev/min*1min/60sec= 480*2pi/60= 50.3 rad/sec
I= 1/2 *3*.12^2=.0216 kg.m^2
KE=1/2 *I*w^2
a uniform disk of mass 3kg and radius 12cm rotates at 480 rev/min. find its kinetic energy (Hint : I of uniform disk =1/2MR2)
To find the kinetic energy of the uniform disk, we first need to calculate its moment of inertia.
Given:
Mass of the disk (m) = 3 kg
Radius of the disk (R) = 12 cm = 0.12 m
Angular velocity (ω) = 480 rev/min
The moment of inertia (I) of the disk can be calculated using the formula:
I = (1/2) * m * R^2
Plugging in the values:
I = (1/2) * 3 kg * (0.12 m)^2
I = (1/2) * 3 kg * 0.0144 m^2
I = 0.0216 kg m^2
Now, we can calculate the kinetic energy (KE) using the formula:
KE = (1/2) * I * ω^2
Plugging in the values:
KE = (1/2) * 0.0216 kg m^2 * (480 rev/min)^2
KE = (1/2) * 0.0216 kg m^2 * (480 rev/min)^2 * (2π rad/rev)^2 * (1 min/60 s)^2
KE = (1/2) * 0.0216 kg m^2 * (480*2π)^2 rev^2/min^2 * (1/3600) min^2/s^2
KE = (1/2) * 0.0216 kg m^2 * (480*2π)^2 rev^2/(60*60*1000) s^2
KE = 0.5 * 0.0216 kg m^2 * 144^2 * π^2 rev^2/(3600*1000) s^2
KE ≈ 1.29 Joules
Therefore, the kinetic energy of the uniform disk is approximately 1.29 Joules.
To find the kinetic energy of a rotating uniform disk, we need to use the formula for rotational kinetic energy, which is given by:
Kinetic Energy (K.E.) = (1/2) * I * w^2
Where:
- K.E. is the kinetic energy
- I is the moment of inertia of the disk
- w is the angular velocity of the disk
From the hint given, we know that for a uniform disk, the moment of inertia (I) is equal to (1/2) * m * R^2, where m is the mass of the disk and R is the radius.
Given values:
Mass (m) = 3 kg
Radius (R) = 12 cm = 0.12 m
Angular velocity (w) = 480 rev/min
To use the formula, we need to convert the angular velocity from rev/min to rad/s. We know that 1 revolution is equal to 2π radians. Thus, we can use the following conversion:
1 rev/min = (2π rad/1 rev) * (1 min/60 s) = (π/30) rad/s
Now we can calculate the angular velocity (w) in rad/s:
w = (π/30) rad/s * 480 = 16π rad/s
Next, we can substitute the values into the formula:
K.E. = (1/2) * I * w^2
K.E. = (1/2) * ((1/2) * m * R^2) * (w^2)
K.E. = (1/2) * ((1/2) * 3 kg * (0.12 m)^2) * (16π rad/s)^2
Simplifying the equation:
K.E. = (1/2) * (1/4) * 3 * (0.12)^2 * (16π)^2
K.E. = (9/64) * (0.12)^2 * (16π)^2
Using a calculator to evaluate the expression, we find that the kinetic energy of the rotating disk is approximately:
K.E. ≈ 11.34 J
Therefore, the kinetic energy of the uniform disk is approximately 11.34 Joules.