A 4.0 kg pulley of radius 0.15 m is pivoted about an axis through its center. What constant torque is required for the pulley to reach an angular speed of 25 rad/s after rotating 6.0 revolutions, starting from rest?
Torque=momentinertia*anglular acceleration
Torque= I * angacc
but
wf^2=wi^2+2*angacc*displacement
wf^2=wi^2+2*angacc*revolutions*2PI
where w is in rad/sec, angacc in rad/sec^2
solve for angacc, then torque
To find the torque required for the pulley to reach the given angular speed, we can use the principle of conservation of angular momentum.
Angular momentum (L) is defined as the product of the moment of inertia (I) and the angular velocity (ω).
L = I * ω
The moment of inertia of a pulley can be calculated using the formula:
I = 0.5 * m * r^2
where m is the mass of the pulley and r is its radius.
In this case, the pulley has a mass of 4.0 kg and a radius of 0.15 m. Substituting these values into the formula, we can find the moment of inertia:
I = 0.5 * 4.0 kg * (0.15 m)^2 = 0.09 kg*m^2
Now, let's calculate the initial angular momentum (L_initial) of the pulley when it is at rest. Since it starts from rest, its initial angular velocity (ω_initial) will be zero.
L_initial = I * ω_initial = 0.09 kg*m^2 * 0 = 0
The final angular momentum (L_final) of the pulley can be calculated using the formula:
L_final = I * ω_final
where ω_final is the desired final angular velocity of 25 rad/s. The pulley completes 6.0 revolutions, which is equivalent to 6 * 2π radians. Therefore, the final angular velocity can be calculated using the formula:
ω_final = (2π * 6.0 revolutions) / t
where t is the time taken to reach the final angular velocity.
Simplifying the equation:
25 rad/s = (2π * 6.0 revolutions) / t
Solving for t:
t = (2π * 6.0 revolutions) / 25 rad/s
Now, we can substitute the value of t back into the formula to calculate L_final:
L_final = I * ω_final = 0.09 kg*m^2 * 25 rad/s
Finally, we can calculate the torque (τ) required to achieve the desired angular velocity change:
τ = (L_final - L_initial) / t
Substituting the values:
τ = (0.09 kg*m^2 * 25 rad/s - 0) / [(2π * 6.0 revolutions) / 25 rad/s]
Simplifying the equation gives the constant torque required to achieve the desired angular speed.