# Rotational Motion - Understanding the Idea

• Arman777
In summary, the use of the ##\vec \theta## unit vector simplifies rotational motion, particularly in cases of uniform circular motion. The formula for acceleration, ##\vec a = R\alpha \vec \theta + R\omega^2 (-\vec r)##, is a general formula and does not necessarily require a force to be applied for uniform circular motion. The value of ##\vec a_t## is determined by the applied tangential force, not the other way round.
Arman777
Gold Member
1- I am trying to understand the rotational motion and In most books there's ##\vec θ## unit vector which its, ##\vec θ=(-sinθ,cosθ)##
I can see that If ##\vec r## unit vector is,##\vec r=(cosθ,sinθ)## then ,

##\frac {d\vec r} {dt}=\vec θ##

Like calculating ##\vec v=\frac {d\vec r} {dt}=R.\frac {dθ} {dt}.\vec θ##

I don't understand why we use this ? Just to make things easier ?

2-And we know that ##\vec a=R∝\vec θ+Rw^2(-\vec r)##
here
##\vec {a_t}=R∝\vec θ## and ##\vec {a_r}=Rw^2(-\vec r)##

I know that If ##a_t## is zero then it becomes uniform circular motion.But In main equations there is an non-zero angular acceleration.So it means do we have to exert ##\vec F=m(-R∝\vec θ)## to create a uniform circular motion ?

Note:This is not a homework question so,I didnt ask there

Arman777 said:
1- I am trying to understand the rotational motion and In most books there's ##\vec θ## unit vector which its, ##\vec θ=(-sinθ,cosθ)##
I can see that If ##\vec r## unit vector is,##\vec r=(cosθ,sinθ)## then ,

##\frac {d\vec r} {dt}=\vec θ##

Like calculating ##\vec v=\frac {d\vec r} {dt}=R.\frac {dθ} {dt}.\vec θ##

I don't understand why we use this ? Just to make things easier ?
In general yes. It simplifies rotational motion a lot. For instance, consider uniform circular motion. If you were to stick to ##x## and ##y## coordinates, they would be constantly changing and they become difficult to work with. They are best suited to translational motion because of this. If you use ##r## and ##\theta## coordinates you can immediately see that one derivative is zero, and that the other is simply the angular speed. The reason we use ##\vec \theta## is to give that speed a direction.

Arman777 said:
2-And we know that ##\vec a=R∝\vec θ+Rw^2(-\vec r)##
here
##\vec {a_t}=R∝\vec θ## and ##\vec {a_r}=Rw^2(-\vec r)##

I know that If ##a_t## is zero then it becomes uniform circular motion.But In main equations there is an non-zero angular acceleration.So it means do we have to exert ##\vec F=m(-R∝\vec θ)## to create a uniform circular motion ?

Note:This is not a homework question so,I didnt ask there
No we do need to apply such a force for uniform circular motion. The ##\vec {a_t}## has to be caused by a force. If there is no tangential force, then it is zero. The term appears in the formula because the formula is general, so it works for any value of ##\vec {a_t}##, including zero. You obtain the value of ##\vec {a_t}## from the value of ##\vec F_t##, not the other way round.

Arman777
Albertrichardf said:
In general yes. It simplifies rotational motion a lot. For instance, consider uniform circular motion. If you were to stick to ##x## and ##y## coordinates, they would be constantly changing and they become difficult to work with. They are best suited to translational motion because of this. If you use ##r## and ##\theta## coordinates you can immediately see that one derivative is zero, and that the other is simply the angular speed. The reason we use ##\vec \theta## is to give that speed a direction.No we do need to apply such a force for uniform circular motion. The ##\vec {a_t}## has to be caused by a force. If there is no tangential force, then it is zero. The term appears in the formula because the formula is general, so it works for any value of ##\vec {a_t}##, including zero. You obtain the value of ##\vec {a_t}## from the value of ##\vec F_t##, not the other way round.

I understand now.Thanks

Welcome.

## 1. What is rotational motion?

Rotational motion is the movement of an object around an axis or center point. This type of motion is often seen in objects such as spinning tops, wheels, and planets orbiting around a star.

## 2. What is the difference between linear and rotational motion?

The main difference between linear and rotational motion is the type of path followed by the object. Linear motion involves movement in a straight line, while rotational motion involves movement along a circular path around an axis.

## 3. How is rotational motion measured?

Rotational motion is often measured using angular velocity, which is the rate of change of angular displacement. It is usually measured in radians per second (rad/s) or degrees per second (deg/s).

## 4. What is torque in rotational motion?

Torque is the measurement of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to an object by the distance from the axis of rotation to the point where the force is applied.

## 5. What are some real-world examples of rotational motion?

Some common examples of rotational motion in everyday life include the rotation of a bike wheel, the spinning of a ceiling fan, and the movement of gears in a car's transmission. Additionally, the rotation of planets and galaxies in space is also a form of rotational motion.

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