How Torque and Speed Affect DC Motor Performance
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 Understanding the Relation Between Torque and Speed
1. What is the fundamental relationship between torque and speed?
The fundamental relationship is that torque is inversely proportional to speed, assuming constant power. This means that for a given amount of power output (like from an engine or motor), an increase in rotational speed (like RPM) will result in a decrease in torque, and vice versa. High torque is achieved at low speeds, and high speed is achieved at low torque.
2. What is the formula that connects power, torque, and speed?
The formula connecting power (P), torque (τ), and angular speed (ω) is P = τ × ω. In this equation, Power is typically measured in Watts, Torque in Newton-meters (N·m), and angular speed in radians per second. This equation is crucial for understanding how mechanical systems transmit energy and is a core concept in rotational dynamics.
3. How do torque and force differ in physics?
While related, torque and force describe different actions. The main differences are:
- Nature of Motion: Force causes an object to undergo linear acceleration (move in a straight line). Torque causes an object to undergo angular acceleration (rotate or twist around an axis).
- Application: Force is a direct push or pull. Torque is a turning or twisting effect of a force applied at a distance from an axis of rotation.
- Vector Type: Force is a true vector, whereas torque is technically a pseudovector (or axial vector), whose direction is determined by the right-hand rule.
4. What is the SI unit of torque, and why isn't it measured in Joules?
The SI unit of torque is the Newton-meter (N·m). Although dimensionally equivalent to the Joule (1 J = 1 N·m), torque is not measured in Joules because they represent different physical quantities. A Joule is a unit of energy or work, which are scalar quantities. Torque, however, is a vector quantity representing a turning effect. Using 'N·m' exclusively for torque prevents confusion between the turning action of torque and the energy transferred as work.
5. How does a car's gearbox demonstrate the relationship between torque and speed?
A car's gearbox is a perfect real-world example of the torque-speed relationship. When starting, the car uses a low gear. This provides high torque and low speed, which is necessary to overcome inertia and get the car moving. Once the car is moving, the driver shifts to a higher gear. This reduces the torque but allows for higher speeds, which is more efficient for cruising. Each gear change modifies the ratio to trade torque for speed or vice versa, based on a relatively constant engine power.
6. If torque and speed are inversely related, does a motor with high torque always have low speed?
Not necessarily. The inverse relationship between torque and speed exists for a constant power output. A motor can be designed to have both high torque and high speed, but this requires a significantly higher power rating. For example, a high-performance electric vehicle motor delivers both high torque for acceleration and high speed for top-end performance because it is designed with a very high power capacity (P = τ × ω). The trade-off exists within the operating limits of a *specific* motor, not as a universal limitation across all motors.
7. Why is the concept of torque important in understanding rotational motion?
Torque is fundamentally important because it is the rotational equivalent of linear force. Just as an unbalanced force causes a change in linear motion (acceleration), an unbalanced torque causes a change in rotational motion (angular acceleration). Without understanding torque, we cannot analyse why objects start to spin, stop spinning, or change their rate of rotation. It is the key to understanding everything from how a wrench tightens a bolt to the dynamics of planetary orbits and the functioning of electric motors.
8. What is the relationship between torque and angular momentum?
The relationship is that torque is the rate of change of angular momentum, just as force is the rate of change of linear momentum. This is expressed by the formula τ = dL/dt, where τ is the net external torque and L is the angular momentum. This principle explains the conservation of angular momentum: if no net external torque acts on a system (τ = 0), its angular momentum remains constant. This is why an ice skater spins faster when they pull their arms in—they reduce their moment of inertia, and to conserve angular momentum, their angular speed must increase.
