A phonograph record of radius 6 cm and mass 15 g is rotating with an angular velocity of 33.33 rev/min. A Penny of mass 1 g suddenly falls on the record and lands at the radial distance of 4 cm from the record’s center. What is the angular velocity after the collision.
The turntable will not slow down if the power is on. A motor keeps it moving at 33.33 rpm. Records (LP) did not have 6 cm radius; it was 6 inches.
If the record player were "coasting" and under no power, you would still need to know the mass of the turntable platen under the record, if you want to predict final angular velocity.
They may want to to apply conservation of angular momentum, but this is a poor choice of problem for applying it.
To determine the angular velocity after the collision, we can use the principle of conservation of angular momentum.
The angular momentum of the rotating phonograph record before the collision can be calculated as the product of its moment of inertia and angular velocity. The moment of inertia of a solid disk rotating about its center is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass, and r is the radius of the disk.
Given that the radius of the phonograph record is 6 cm and the mass is 15 g, the moment of inertia (I1) before the collision can be calculated as:
I1 = (1/2) * 15 g * (6 cm)^2
The angular velocity before the collision is given as 33.33 rev/min. To convert this to radians per second, we can use the conversion factor:
1 rev/min = (2π/60) rad/s
Therefore, the initial angular velocity (ω1) is:
ω1 = 33.33 rev/min * (2π/60) rad/s
Next, the penny falls onto the phonograph record at a radial distance of 4 cm from the center. To calculate the moment of inertia after the collision (I2), we need to consider the additional mass of the penny.
The moment of inertia of a point mass rotating about an axis passing through its center is given by the formula:
I = m * r^2
where m is the mass, and r is the radial distance from the axis of rotation.
Given that the mass of the penny is 1 g and its radial distance from the center is 4 cm, the moment of inertia (I2) after the collision can be calculated as:
I2 = 1 g * (4 cm)^2
Since the angular momentum is conserved during the collision, we can equate the initial angular momentum (I1 * ω1) to the final angular momentum (I2 * ω2).
I1 * ω1 = I2 * ω2
Solving for ω2, we can find the angular velocity after the collision.
ω2 = (I1 * ω1) / I2
Substituting the values, we can calculate the angular velocity after the collision.