Please help me understand with steps, I really don't get it
A square plate has a side length of 1.20 m. A 24.0-N ~ m torque is applied about an axis perpendicular to the plate through its center. If it experiences an angular acceleration of 6.25 rad/s2, calculate the mass of the plate.
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I = (1/6) m L^2
Torque = I * angular acceleration
24 = (1/6) m L^2 * 6.25^2
but L = 1.2
so
24 = (1/6) m (1.2*6.25)^2
To calculate the mass of the plate, we can use the relationship between torque (τ), moment of inertia (I), angular acceleration (α), and mass (m). The moment of inertia of a square plate about an axis perpendicular to it through its center is given by the equation:
I = 1/6 * m * a^2
where m is the mass of the plate and a is the side length of the square plate.
Step 1: Find the moment of inertia (I)
Using the given side length of the square plate (a = 1.20 m), we can substitute this value into the equation:
I = 1/6 * m * (1.20 m)^2
Step 2: Use the torque and angular acceleration
Since torque (τ) is equal to the moment of inertia (I) multiplied by the angular acceleration (α), we can set up the equation:
τ = I * α
Step 3: Substitute the given values
Substitute the given torque (τ = 24.0 N·m) and angular acceleration (α = 6.25 rad/s^2) into the equation:
24.0 N·m = I * 6.25 rad/s^2
Step 4: Solve for moment of inertia (I)
Rearrange the equation to solve for I:
I = 24.0 N·m / 6.25 rad/s^2
Step 5: Calculate the mass (m)
Now that we have the moment of inertia (I), we can substitute this value back into the equation in Step 1 and solve for the mass (m):
I = 1/6 * m * (1.20 m)^2
24.0 N·m / 6.25 rad/s^2 = 1/6 * m * (1.20 m)^2
Step 6: Solve for mass (m)
To isolate the mass (m), multiply both sides of the equation by 6/1.20^2:
m = (24.0 N·m / 6.25 rad/s^2) * (6 / 1.20^2)
Step 7: Calculate the mass (m)
Evaluate the expression on the right side of the equation to find the mass (m).
By following these steps, you should be able to calculate the mass of the square plate.