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