(Use the fact that the kinetic energy of a particle of mass "m" moving at a speed "v" is (1/2)m(v^2). Slice object into pieces in such a way that the velocity is approximately constant on each piece.)
Q: Find the kinetic energy of a phonograph record of uniform density, mass 50gm and radius 10cm rotating at 33(1/3) revolutions per minute.
To find the kinetic energy of the phonograph record, we need to consider the rotational kinetic energy.
The rotational kinetic energy is given by the formula:
KE_rot = (1/2)Iω²
Where:
KE_rot is the rotational kinetic energy,
I is the moment of inertia of the object, and
ω is the angular velocity.
To determine the moment of inertia, we need to consider how the object is sliced into pieces. Given the given prompt, we can slice the record into thin circular rings or slices, such that the velocity is approximately constant for each slice.
Let's consider a small circular slice of the phonograph record, which has a radius r and a thickness δr. The mass of this slice can be calculated using the density and thickness.
Mass of the slice = density x volume of the slice
= density x (area of the slice x thickness)
= density x (πr²δr)
Since the density is uniform, we can consider it as a constant. So for a given slice, the mass (m) will be equal to:
m = density x (πr²δr)
Now, we need to find the moment of inertia of the slice. The moment of inertia of a circular slice of radius r and thickness δr is given by:
dI = (1/2)m(r²)
Substituting the value of m, we get:
dI = (1/2)(density)(πr²δr)(r²)
Integrating this equation from 0 to R (where R is the radius of the entire record), we can find the total moment of inertia of the record.
I = ∫[0 to R] (1/2)(density)(πr⁴)dr
Now, to find the angular velocity ω, we need to convert the given revolutions per minute (RPM) to radian per second (rad/s).
1 revolution = 2π radians
1 minute = 60 seconds
Angular velocity ω = (2π x 33(1/3)) / 60
Finally, we can substitute the values of I and ω into the formula for rotational kinetic energy, KE_rot:
KE_rot = (1/2)Iω²
Solve this equation to find the kinetic energy of the phonograph record.