A record player is spinning at 32 rpm when the motor is switched off. After 6.8 seconds the angular speed has decreased to 20 rpm. Calculate the frictional torque if the record player can be treated as a solid wheel of mass 2.4 kg and radius 15 cm.
A) 476.47059 Nm
B) 0.00499 Nm
C) 0.01907 Nm
D) 952.94118 Nm
E) 0.04765 Nm
To solve this problem, we can use the principle of conservation of angular momentum.
The angular momentum of an object is defined as the product of its moment of inertia and its angular velocity. In this case, the moment of inertia of the record player can be assumed to be that of a solid wheel, which is given by the equation:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the wheel, and r is the radius of the wheel.
We are given the initial angular velocity (ω1) of the record player when the motor is switched off (32 rpm) and the final angular velocity (ω2) after 6.8 seconds (20 rpm). Note that angular velocity is typically measured in radians per second, so we need to convert the given values to radians per second.
To convert from rpm to radians per second, we can use the conversion factor: 1 rpm = (2π/60) rad/s.
So, ω1 = 32 rpm * (2π/60) rad/s = (32/60) * 2π rad/s
And, ω2 = 20 rpm * (2π/60) rad/s = (20/60) * 2π rad/s
Next, we can use the conservation of angular momentum to relate the initial and final angular velocities with the moment of inertia of the record player:
I1 * ω1 = I2 * ω2
Plugging in the values, we get:
[(1/2) * m * r^2] * [(32/60) * 2π] = [(1/2) * m * r^2] * [(20/60) * 2π]
Notice that the mass and radius cancel out on both sides of the equation. So, we can simplify further:
(32/60) * 2π = (20/60) * 2π
Now, let's solve for the frictional torque (τ).
Frictional torque (τ) is given by the equation:
τ = I * α
where α is the angular acceleration.
We can find α by dividing the change in angular velocity (ω2 - ω1) by the change in time (t).
α = (ω2 - ω1) / t
Plugging in the given values, we get:
α = [(20/60) * 2π - (32/60) * 2π] / 6.8
Simplifying further:
α = (-12/60) * 2π / 6.8
Finally, we can find the frictional torque by multiplying the moment of inertia (I) with the angular acceleration (α):
τ = I * α = [(1/2) * m * r^2] * α
Plugging in the given values, we get:
τ = (1/2) * 2.4 kg * (0.15 m)^2 * [(-12/60) * 2π / 6.8]
Simplifying further:
τ ≈ -0.01907 Nm
Since torque is a scalar quantity, we take the magnitude of the negative value:
τ ≈ 0.01907 Nm
Therefore, the correct answer is C) 0.01907 Nm.