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Relation Between Torque and Speed

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Last updated date: 17th May 2024
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What is Torque?

In physics, you often hear the word torque. Do you know exactly what torque means? Well, a torque is nothing but the force applied on an object to make it rotate about its axis. Therefore, any force that can cause this angular acceleration in an object is torque. 

 

As you can probably understand from this definition, torque is a vector quantity because both the magnitude and direction of a force are at work here. The torque vector direction depends on the direction of the force on the axis. 

 

A common question that can arise when learning about torque is its relation to speed. Therefore, listed below is the equation that pits torque vs speed.

 

\[\text{Torque}(\tau) = \frac{Power}{speed}\]

 

This is the most basic form of a relation between torque and speed. If you desire to know more, you must first determine what speed is. 

 

Quick Exercise - 1

Q. A wheel moves at a rate of 0.3 m/s on applying a power of 40 Watts. Determine the torque acting on the wheel using a torque-speed relation equation.

Solution –

Speed of the wheel is 0.3 m/s

Power applied on the wheel is 40 watts

Thus, \[Torque = \frac{Power}{speed}\]

\[Torque = \frac{40}{0.3} \]

Torque is equal to 133.33 Newton-metre. 

 

What is Speed?

To assess torque vs speed truly, you must first understand speed in detail. Speed is nothing but the distance traveled by an object per unit of time. Speed is a scalar quantity. You do not need to establish the direction of movement to determine the speed of a body.

 

This is also what sets speed apart from velocity. Being a vector quantity, velocity is the speed of a body in a particular direction. 

 

Torque and Speed Formula

Both of the relationship between these two and their formula can be easily understood by the formula to calculate the power carried by an object moving in a circular motion.

 

\[Power = Torque \times Speed\]

 

\[P = \tau \times \omega \]

 

Where p represents the power or the work done by the object in a circular motion. T is the torque (Torque is considered as the rotational ability of a body, considered like the equivalent of force) and \[\omega\] is the angular speed or velocity attained by the moving object (considered as the rate of change in angular displacement).

 

The equation above can be rearranged to get the formula to find the torque on  the moving object

 

\[\tau = \frac{P}{\omega}\]

 

Similarly, the formula to calculate the speed (Angular speed/ velocity)

 

\[\omega= \frac{P}{T} \]

 

The force in here is typically measured in Watts (W) or horsepower (hp). In motors, it is basically the mechanical output power of a motor. For electric motors the speed is measured in the revolutions per minute, or RPM, It defiance the rate of rotation of the moving part. Torque for electrical appliances is measured in either inch pounds (in lbs) or Newton metres (N m) and it defines the force exerted on the motor or the other object in circular motion. It is the rotational force that the object deleveloped.

 

Deriving the Relation Between Torque and Speed Formula

Since torque is a rotatory motion, we can easily derive its relation to power by comparing the linear equivalent. To determine the linear displacement, simply multiply the radius of movement with the angle covered. Keep in mind that linear displacement refers to the distance covered at the circumference of a wheel.

 

Therefore, we can say that \[\text{Linear distance} = Time \times \text{Angular velocity} \times Radius\] (eq.1) 

 

We know that \[Torque = Force \times Radius \]

 

\[Force = \frac{Torque}{Radius}\] (eq.2)

 

Now, \[Power = \frac{Force \times \text{Linear Distance}}{Time} \]

 

Integrating the value of force from eq.1 and eq.2, we get

 

\[Power = \frac{(\frac{Torque}{Radius}) \times Time \times \text{Angular Velocity} \times Radius}{Time} \]

 

Thus, \[Power = Torque \times \text{Angular Velocity}\]

 

Consequently, \[Torque = \frac{Power}{\text{Angular velocity}}\]

 

What is the Relationship Between Torque and Speed?

The mathematical formula tells us that this force around an axis is inversely proportional to speed (angular velocity). This means that an increase in velocity causes torque to drop and vice versa.

 

Another vital factor that you need to keep in mind is that in this equation velocity and speed is used interchangeably. This is because torque, being a vector quantity, will always have speed in a particular direction. Now that you know the torque and speed relation, answer this simple question.

 

True or False – 1

Q. Torque is directly proportional to the radius of rotation.

Ans. True. Since torque is the product of force and radius of rotation, increasing this radius will also increase the resulting torque. The same is true for the opposite as well.

 

Relation Between Torque and Speed in DC Motor

In a DC motor, speed is calculated in the form of rotation per minute. Thus for such a motor, you can determine torque, by using the following formula – 

 

\[Torque = \frac{Power}{(2\pi \times \text{Speed of Rotation})}\]

 

Our online classes and a wide selection of PDF books will help you further your understanding of torque vs speed. We also have doubt-clearing sessions to ensure proper comprehension of each topic. Now, you can even download our Vedantu app to access online sessions with ease.

FAQs on Relation Between Torque and Speed

1. What is the SI Unit for Measuring Torque?

The SI unit of torque is Newton-metre. The quantity torque (or moment of force) can be thought of as the cross product of force and distance, the dimension of the torque will be N m. And the dimension of the Newton is kg m s-2.


Hence the final dimensions of the quantity torque is kg m2 s-2. As one can observe that these are the dimensions of the work done and are equal to the Joule, But the Torque is not the not work or energy, this can’t be expressed in Joules (J), and can only be defined by the use of Newton meter.

2. What is the Difference Between Torque and Force?

Torque is a force applied along an angular direction or the force which results in the rotation of the object. Force, on the other hand, is any push or pull that causes a change in acceleration of the object. Force is a true vector quantity with the S.I. unit of Newton, while the torque is a pseudovector quantity and as it is the product of force and distance, the S.I. unit of the torque is equivalent to the work done unit (J) which is kg m2 s-2.  The derivative of the force results in the linear momentum while the derivative of the torque gives the angular momentum.

3. If I Increase the Speed, will the Torque on the Object Increase or Decrease?

Torque is inversely proportional to speed. Thus, when speed increases, torque will decrease. T can understand via the formula if the torque


\[\frac{P}{\omega}\] 


Simply, if the angular velocity (or speed) will increase the torques on a certain object, the torque on that object will have to decrease in order to balance out the equation for the power consumed. If one wants to maintain the same amount of torque then we will need to at least increase the power of that certain object.

4. What is torque and is it a vector?

The concept of torque originates from the studies by Archimedes on the usage of levers. The torque is defined as the force that when applied results in the rotation of the moving object, it is the rotational equivalent of the linear force and is known by many names according to the various branches of physics. These names are moment, a moment of force, rotational force or turning effect. The formula to find it is written as below


\[\tau = r \times F \Rightarrow \tau = |r| |F| cos\] 


The torque does seem like a vector quantity as it is the product of two vectors and it is a vector quantity, but in certain circumstances, the torque is only represented as a scalar quantity, hence due to this reason torque is identified as a Pseudovector quantity.

5. What is the relationship between torque and speed in a motor?

By the formula of calculating the power, we can easily say that there exists an inversely proportional relationship between two quantities, torque and speed.


The formula for finding the power

\[Power = Torque \times Speed\] 


\[P = \tau \times \omega  \]


This explains that to maintain the equation, if torque increases then the speed had to decrease and when speed got increased, then torque must decrease. 


Students, please note that in the equation above the angular speed and angular velocity, both can be used interchangeably.