# Why Does ɪa = TR in Angular Motion?

• Luke0034
In summary, the formula ɪa = TR represents the rotational analog of Newton's second law of motion, where torque (TR) is equal to the moment of inertia (ɪ) multiplied by the angular acceleration (a). This can be derived from the fact that torque is equal to force multiplied by radial distance, and the tension (T) in the rope is the force applied to the drum at a distance (R) from the axis of rotation. This formula can be used to solve problems involving rotational motion, such as the example of a stuntman jumping off a building with a rope tied around his waist and connected to a cylindrical drum with rope wound around it.
Luke0034

## Homework Statement

I am first learning about angular motion and came across this formula while doing a homework problem. Can anyone explain to me why ɪa = TR? That is the moment of inertia * angular acceleration = tension in rope * radius.

For reference, in the problem it's a stunt guy jumping off a building with rope tied around his waist connected to a cylindrical drum with rope wound around it.

I'm only interested in knowing why those two quantities equal each other. Thanks in advance!

torque = ɪ

torque = TR

ɪa = TR

## The Attempt at a Solution

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I already solved the problem, just need clarification.

TR is the torque exerted on the drum by the rope. Do you know Newton's second law of motion:

Force = mass x acceleration

Well its rotational analog is:

Torque = Moment of inertia x angular acceleration.

Each item in that equation is the angular analog of the corresponding element in the Newton law. The second equation can be derived from the first.

andrewkirk said:
Well its rotational analog is:

Torque = Moment of inertia x angular acceleration.

Just for info all the equations of motion (for example SUVAT) have a rotational analog.

Hey, I'm also working on rotational motion in physics right now as well. Whenever I try and fit things into equations I think of the units involved.

I = MR^2 = Kg * m^2

T = N* Meters = Kg * m^2/(s^2)

I = Ta

Kg * m^2 = (Kg * m^2/(s^2)) * rad/ s^2

radian has no dimensions so the left side and right side of the above equation end up being the same.Basically do the same with the given equation and you should see why the equation works out.

andrewkirk said:
TR is the torque exerted on the drum by the rope. Do you know Newton's second law of motion:

Force = mass x acceleration

Well its rotational analog is:

Torque = Moment of inertia x angular acceleration.

Each item in that equation is the angular analog of the corresponding element in the Newton law. The second equation can be derived from the first.
I understand that T = Ia is the rotational analog of F = ma, but why does the tension * radius = Ia? Why does TR = Ia... why are both Ia and TR both equal to torque?

Did you read your textbook? It's very likely explained in there. Here's a derivation for a point mass.
\begin{align*}
\vec{F} &= m\vec{a} \\
\vec{r}\times\vec{F} &= m(\vec{r}\times\vec{a}) \\
\lvert \vec{r}\times\vec{F} \rvert &= m\lvert \vec{r}\times\vec{a} \rvert \\
rF_\perp &= mra_\perp \\
rF_\perp &= mr(r\alpha) \\
rF_\perp &= mr^2 \alpha \\
rF_\perp &= I\alpha
\end{align*}

conscience
Luke0034 said:
I understand that T = Ia is the rotational analog of F = ma, but why does the tension * radius = Ia? Why does TR = Ia... why are both Ia and TR both equal to torque?

In plain English... Torque equals force multiplied by radial distance.

Depending on the problem the force might be provided by "tension" and the distance might be the "radius". However this isn't always the case.

It sounds like you are working on a particular problem/example. I suggest you post the details.

Luke, I think a picture would be very helpful. However from the description I suspect this question is all about the drum. The radius R is the radius of the drum. The rope comes off of the drum at a tangent. The tension on the rope is also the force the rope is applying to the drum. Force applied tangentially at a distance from the axis of rotation is the definition of torque. The rope applies a torque TR to the drum.

Cutter Ketch said:
Luke, I think a picture would be very helpful. However from the description I suspect this question is all about the drum. The radius R is the radius of the drum. The rope comes off of the drum at a tangent. The tension on the rope is also the force the rope is applying to the drum. Force applied tangentially at a distance from the axis of rotation is the definition of torque. The rope applies a torque TR to the drum.

Okay wow that makes a lot more sense. I didn't know that was the exact definition. I thought it was just some sort of angular force. Makes a lot more sense now, I understand. Thanks a lot!

Luke0034 said:
Okay wow that makes a lot more sense. I didn't know that was the exact definition. I thought it was just some sort of angular force. Makes a lot more sense now, I understand. Thanks a lot!
You're welcome

## 1. Why is ɪa equal to TR in Angular Motion?

The equation ɪa = TR in Angular Motion represents the relationship between the moment of inertia (ɪ) and the angular acceleration (a) of an object. This equation is derived from Newton's Second Law of Motion, which states that the net torque (T) on an object is equal to its moment of inertia multiplied by its angular acceleration. Therefore, ɪa = TR is a fundamental equation in understanding the rotational motion of objects.

## 2. How is the moment of inertia related to angular acceleration in Angular Motion?

The moment of inertia (ɪ) is a measure of an object's resistance to changes in its rotational motion. It is directly proportional to the angular acceleration (a) of the object, meaning that as the moment of inertia increases, the angular acceleration decreases, and vice versa. This relationship is represented by the equation ɪa = TR, where T is the net torque acting on the object.

## 3. Is the equation ɪa = TR applicable to all objects in Angular Motion?

Yes, the equation ɪa = TR is applicable to all objects in Angular Motion, regardless of their shape or size. This is because the moment of inertia (ɪ) takes into account the distribution of mass within an object, making it a comprehensive measure of an object's resistance to changes in its rotational motion. Therefore, the equation holds true for all objects experiencing angular motion.

## 4. Can ɪa = TR be used to calculate the angular acceleration of an object?

Yes, the equation ɪa = TR can be rearranged to solve for the angular acceleration (a) of an object. This can be useful in predicting the behavior of an object undergoing rotational motion, as well as in designing and analyzing machines and systems that involve rotational motion. However, it is important to note that this equation is only valid for objects with a constant net torque acting on them.

## 5. How does understanding ɪa = TR in Angular Motion benefit scientific research?

Understanding the equation ɪa = TR in Angular Motion has many practical applications in scientific research. It allows scientists to accurately predict and analyze the rotational behavior of objects, which is crucial in fields such as engineering, physics, and biomechanics. This equation also provides a fundamental understanding of rotational motion, which can be applied to a wide range of real-world problems and innovations.

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