A solid cylindrical disk has a radius of 0.18 m. It is mounted to an axle that is perpendicular to the circular end of the disk at its center. When a 40-N force is applied tangentially to the disk, perpendicular to the radius, the disk acquires an angular acceleration of 90 rad/s2. What is the mass of the disk?
thanks jews
To determine the mass of the disk, we'll need to use the following formula:
Torque = Moment of Inertia * Angular acceleration
First, let's calculate the moment of inertia of the disk. The moment of inertia for a solid circular disk rotating about an axle through its center is given by the formula:
I = (1/2) * m * r^2
Where:
I = Moment of Inertia
m = Mass of the disk
r = Radius of the disk
Given:
r = 0.18 m
Angular acceleration = 90 rad/s^2
Rearranging the formula, we have:
I = (2 * Torque) / (Angular acceleration)
The torque is given by the formula:
Torque = Force * Radius
Given:
Force = 40 N
Substituting the values, we have:
I = (2 * (40 N * 0.18 m)) / 90 rad/s^2
I = (80 N * 0.18 m) / 90 rad/s^2
I = 0.36 N*m*s^2 / rad
Now, let's solve for the mass of the disk. Rearranging the formula for the moment of inertia, we have:
m = (2 * I) / (r^2)
Substituting the values, we have:
m = (2 * 0.36 N*m*s^2 / rad) / (0.18 m)^2
m = (0.72 N*m*s^2 / rad) / (0.0324 m^2)
m ≈ 22.22 kg
Therefore, the mass of the disk is approximately 22.22 kg.
To find the mass of the disk, we can use the formula:
Torque = Moment of inertia * Angular acceleration
The torque can be calculated by multiplying the applied force by the radius of the disk. The moment of inertia for a solid disk rotating about an axis perpendicular to its center can be calculated using the formula:
Moment of inertia = (1/2) * mass * radius^2
Here, we know the torque (40 N * 0.18 m) and the angular acceleration (90 rad/s^2). By rearranging the formulas, we can solve for the mass of the disk.
First, calculate the torque:
Torque = Force * Radius
Torque = 40 N * 0.18 m
Torque = 7.2 Nm
Next, rearrange the formula for torque to solve for the moment of inertia:
Torque = Moment of inertia * Angular acceleration
7.2 Nm = Moment of inertia * 90 rad/s^2
Now, substitute the moment of inertia formula and solve for the mass:
7.2 Nm = (1/2) * mass * radius^2 * 90 rad/s^2
Simplify the equation:
mass * radius^2 = (2 * 7.2 Nm) / (90 rad/s^2)
mass * (0.18 m)^2 = 0.16 kg
mass = 0.16 kg / (0.18 m)^2
Calculating the mass:
mass = 0.16 kg / (0.18 m)^2
mass = 0.16 kg / 0.0324 m^2
mass ≈ 4.94 kg
Therefore, the mass of the disk is approximately 4.94 kg.