A solid cylindrical disk has a radius of 0.16 m. It is mounted to an axle that is perpendicular to the circular end of the disk at its center. When a 50 N force is applied tangentially to the disk, perpendicular to the radius, the disk acquires an angular acceleration of 120 rad/s2. What is the mass of the disk?
Determine I from the angaccleration and torque. From I, you can determine the mass.
To find the mass of the disk, we need to determine its moment of inertia (I) first. The moment of inertia is a measure of an object's resistance to changes in its rotational motion.
We can use the formula for torque (τ) to find the moment of inertia:
τ = I * α
Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
In this case, the torque (τ) is equal to the product of the applied force (F) and the radius (r):
τ = F * r
Now we can substitute this expression for torque (τ) into the previous equation:
F * r = I * α
Rearranging the equation, we get:
I = (F * r) / α
Now we know the values of the applied force (F = 50 N), the radius (r = 0.16 m), and the angular acceleration (α = 120 rad/s^2). Substituting these values in:
I = (50 N * 0.16 m) / 120 rad/s^2
Simplifying the equation, we get:
I = 0.0667 N * m * s^2
The moment of inertia (I) is measured in kilogram-meter squared (kg * m^2), so we need to convert it to the appropriate unit.
Finally, we can use the formula for the moment of inertia of a solid cylinder:
I = (1/2) * m * r^2
We want to find the mass (m), so we can rearrange the formula:
m = (2 * I) / r^2
Plugging in the values for I (0.0667 N * m * s^2) and r (0.16 m), we can calculate the mass:
m = (2 * 0.0667 N * m * s^2) / (0.16 m)^2
m = 8.34 kg
Therefore, the mass of the disk is approximately 8.34 kg.