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.
moment of inertia: mL^2/6=m/6*1.2^2 = m*1.44/6
torque=Moment *angular acceleration
24=m*1.44/6*6.25 solve for mass m.
To calculate the mass of the plate, we need to use the formula for torque and moment of inertia.
The formula for torque is given by:
τ = Iα
Where,
τ = Torque (Nm)
I = Moment of inertia (kg·m²)
α = Angular acceleration (rad/s²)
First, let's calculate the moment of inertia of the square plate. For a square plate, the moment of inertia can be calculated using the formula:
I = (1/6) × m × (a^2 + b^2)
Where,
m = Mass of the plate (kg)
a = Side length of the plate (m)
b = Side length of the plate (m)
Given:
a = 1.20 m
b = 1.20 m
τ = 24.0 N·m
α = 6.25 rad/s²
We have τ and α, and we need to find m.
First, let's calculate the moment of inertia:
I = (1/6) × m × (a^2 + b^2)
Plugging in the values:
I = (1/6) × m × (1.20^2 + 1.20^2)
Simplifying:
I = (1/6) × m × (2.88 + 2.88)
I = (1/6) × m × (5.76)
Now, let's substitute the value of τ and α into the torque formula:
τ = Iα
24.0 = (1/6) × m × (5.76) × 6.25
Simplifying:
24.0 = (1/6) × m × (5.76 × 6.25)
24.0 = (1/6) × m × 36
Divide both sides by (1/6) × 36:
24.0 / [(1/6) × 36] = m
m = 2.0 kg
Therefore, the mass of the square plate is 2.0 kg.