[Physics Class Notes] on Relation Between Torque and Speed Pdf for Exam

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}{(2pi times text{Speed of Rotation})}]

 

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