An electric motor exert a constant torque of 10Nm on a grindstone about the shaft is 2.0kg/m². If the system starts from rest, find the work done by the motor in 8sec.
To find the work done by the motor in 8 seconds, we need to calculate the rotational kinetic energy (work) of the grindstone.
The rotational kinetic energy (KE) of an object is given by the formula:
KE = (1/2) * I * ω²,
where I is the moment of inertia of the object and ω is the angular velocity.
Given:
Torque, τ = 10 Nm
Moment of inertia, I = 2.0 kg/m²
Time, t = 8 s
First, we need to find the angular acceleration (α) using the torque formula:
τ = I * α
Rearranging the formula, we get:
α = τ / I
Substituting the given values, we have:
α = 10 Nm / 2.0 kg/m²
Solving this, we find α = 5 rad/s².
Next, we find the angular velocity (ω) using the formula:
ω = α * t
Substituting the values, we have:
ω = 5 rad/s² * 8 s
Solving this, we find ω = 40 rad/s.
Finally, we can calculate the rotational kinetic energy (work) using the formula:
KE = (1/2) * I * ω²
Substituting the values, we have:
KE = (1/2) * 2.0 kg/m² * (40 rad/s)²
Simplifying this, we find KE = 1600 J.
Therefore, the work done by the motor in 8 seconds is 1600 Joules.