Angular Velocity On Pulley After Variable Force

[Jshumate]

Homework Statement

A pulley having a rotational inertia of 1.1 10^-3 kg·m² about its axle and a radius of 23 cm is acted on by a force, applied tangentially at its rim, that varies in time as F = 0.50t + 0.30t², where F is in Newtons and t in seconds. If the pulley was initially at rest, find its angular speed after 4.0 s.

Homework Equations

(I'm having trouble with Latex so excuse my non-greek)
T = AI
(net torque= angular acceleration*rotational inertia)
w = at
(angular velocity = angular acceleration*time)
T = rF
(torque = length*force)

The Attempt at a Solution

w = (T)(t)/I
rFt/I = w

I assume I have to integrate the force function over the period of time. The integral of F(t) from 0 to 4 = 10.4.
(10.4 N)(0.23 m)(4 s)/(1.1 10^-3kg·m²) = w
w = 8698 rad/s

But this is incorrect. I am not sure how to handle this with the variable force being applied.

 

[ehild]

I assume I have to integrate the force function over the period of time. The integral of F(t) from 0 to 4 = 10.4.
(10.4 N)(0.23 m)(4 s)/(1.1 10^-3kg·m²) = w
w = 8698 rad/s

why did you multiplied by the time?

ehild

 

[SammyS]

Homework Statement

A pulley having a rotational inertia of 1.1 10^-3 kg·m² about its axle and a radius of 23 cm is acted on by a force, applied tangentially at its rim, that varies in time as F = 0.50t + 0.30t², where F is in Newtons and t in seconds. If the pulley was initially at rest, find its angular speed after 4.0 s.

Homework Equations

(I'm having trouble with Latex so excuse my non-greek)
T = AI
(net torque= angular acceleration*rotational inertia)
w = at  dw = a·dt You will need to integrate this.
(angular velocity = angular acceleration*time)
T = rF
(torque = length*force)

The Attempt at a Solution

w = (T)(t)/I  dw = ((T)/I)·(dt)  
rFt/I = w  (rF/I)(dt) = dw

I assume I have to integrate the force function over the period of time. The integral of F(t) from 0 to 4 = 10.4.
(10.4 N)(0.23 m)(4 s)/(1.1 10^-3kg·m²) = w
w = 8698 rad/s

But this is incorrect. I am not sure how to handle this with the variable force being applied.

Some corrections in red above.

$$\omega-\omega_0=\frac{r}{I}\ \int_0^{\,4} F(t) dt$$

I don't think this is the integral you did.

 

[Jshumate]

No idea why I multiplied by time, thanks guys. :D

 

[rwisz]

I would approach this problem by first making sure that the units are consistent. The force should be in Newtons and the time in seconds, but the units given in the problem statement are not clear. Once the units are confirmed, I would use the equation T = rF to calculate the torque on the pulley at each time point, and then use the torque to calculate the angular acceleration using the equation T = AI. From there, I would use the equation w = at to calculate the angular velocity at each time point and then use the equation w = w0 + at to find the final angular velocity at t = 4 seconds, where w0 is the initial angular velocity (which is zero in this case).

Alternatively, I could use calculus to solve this problem by integrating the force function over the time interval of 0 to 4 seconds. This would give me the total change in angular momentum, which I could then use to find the final angular velocity using the equation w = L/I, where L is the angular momentum and I is the rotational inertia.

In either approach, it is important to clearly define the units and carefully apply the equations to ensure an accurate solution. Additionally, I would double check my calculations and make sure my assumptions are correct. If I am still having trouble, I would consult with colleagues or reference materials to get further insight and understanding.