# Newton's Second Law for Rotation

• frig0018
In summary: To integrate over t you need alpha as a function of t. They are asking for alpha at t=3 s, which is 420 rad/s^2. I am fairly sure of this because it is the answer in the back of the book. But i am struggling to figure out omega at t=3 s which should be 500 rad/s but i need the work to back it up and I'm not sure which equations to use.
frig0018

## Homework Statement

A pulley, with a rotational inertia of 1.0 x 10^-3 kg*m^2 about its axle and a radius of 10 cm, is acted on tangentially at its rim. the force magnitude varies in time as F=0.50t + 0.30t^2, with F in Newtons and t in seconds. The pulley is initially at rest . At t=3.0 s what are its (a) angular acceleration and (b) its angular speed?

## Homework Equations

(a) $$\alpha$$= (r*F(3)) / I = 420 rad/sec^2
(b) unsure, but tried using $$\omega$$= $$\omega$$$$_{}0$$ +$$\alpha$$t = 1260 rad/s which is wrong and several other variations of that equation.

## The Attempt at a Solution

(see above)

The kinematic equation you used is incorrect because this is only for cases of constant alpha. In this case, since force is changing, torque is changing. And since torque is changing, alpha is changing. Think of the general (geometric?) relationship between alpha and omega.

so is the geometric relationship that when alpha increases, so does omega? do i need to use an integral?
dw= (integral)alpha dt? but how can i differentiate this if there are no variables?
i come up with dw=alpha*t + c but when i plug in t=3, i still get 1260...

Hi frig0018,

alpha is not 420 rad/s^2 except at t=3. To integrate over t you need alpha as a function of (the variable) t.

they are just asking for alpha at t=3 s, which is 420 rad/s^2. I am fairly sure of this because it is the answer in the back of the book... However, i am struggling to figure out omega at t=3 s which should be 500 rad/s but i need the work to back it up and I'm not sure which equations to use.

Your method for obtaining the alpha at t=3 is correct, and your idea of integrating is also correct, it seems you're just stuck on finding the function? Well, look at the way in which you solved for alpha and you'll see you have a general formula for alpha. That should help :)

frig0018 said:
they are just asking for alpha at t=3 s, which is 420 rad/s^2. I am fairly sure of this because it is the answer in the back of the book... However, i am struggling to figure out omega at t=3 s which should be 500 rad/s but i need the work to back it up and I'm not sure which equations to use.

Yes, your answer for part a looks correct, but my previous comment:

alphysicist said:
Hi frig0018,

alpha is not 420 rad/s^2 except at t=3. To integrate over t you need alpha as a function of (the variable) t.

was referring to part b. In post #3 you tried to integrate (alpha dt), with the integral running from t=0 to t=3. But if you integrate from t=0 to t=3, you can't just plug in the value of alpha that is true only at t=3. You need alpha as a function of t.

They give you the force as a function of t: F= 0.50t + 0.30t^2. What is alpha as a function of t?

## 1. How does Newton's Second Law for Rotation differ from his Second Law for linear motion?

While Newton's Second Law for linear motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, his Second Law for Rotation states that the angular acceleration of an object is directly proportional to the net torque acting on it and inversely proportional to its moment of inertia.

## 2. What is the formula for Newton's Second Law for Rotation?

The formula for Newton's Second Law for Rotation is T = Iα, where T is the net torque, I is the moment of inertia, and α is the angular acceleration.

## 3. How is Newton's Second Law for Rotation applied in real-world situations?

Newton's Second Law for Rotation is commonly used in the design and analysis of rotating systems, such as car engines, turbines, and propellers. It is also used in sports, such as figure skating and diving, to understand how angular momentum and torque affect the movement of the athlete.

## 4. Can Newton's Second Law for Rotation be used to calculate the motion of an object in a circular path?

Yes, Newton's Second Law for Rotation can be used to calculate the motion of an object in a circular path, as circular motion can be seen as a type of rotational motion. The net torque acting on the object will cause it to undergo angular acceleration, which will result in circular motion.

## 5. Are there any limitations to Newton's Second Law for Rotation?

Like any scientific law, Newton's Second Law for Rotation has its limitations. It assumes that the object is rigid and that there are no external forces acting on it. In reality, most objects are not completely rigid and there are always external forces present, so this law provides an idealized understanding of rotational motion.

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