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Learn the dimensional formula of torque and its derivation from the web page. Find out the dimensions of mass, length, time and angular velocity and acceleration.
Feb 3, 2024 · Learn what is torque, how to derive its dimensional formula and its advantages. The dimensional formula of torque is [ML2T-2] and it is the same as energy and work.
- Overview
- What is torque?
- How is torque calculated?
- How is torque measured?
- What role does torque play in rotational kinematics?
- What is rotational equilibrium?
- How does torque relate to power and energy?
- How can we increase or decrease torque?
- Data sources
Learn how to find the torque exerted by a force.
Torque is a measure of the force that can cause an object to rotate about an axis. Just as force is what causes an object to accelerate in linear kinematics, torque is what causes an object to acquire angular acceleration.
Torque is a vector quantity. The direction of the torque vector depends on the direction of the force on the axis.
Anyone who has ever opened a door has an intuitive understanding of torque. When a person opens a door, they push on the side of the door farthest from the hinges. Pushing on the side closest to the hinges requires considerably more force. Although the work done is the same in both cases (the larger force would be applied over a smaller distance) people generally prefer to apply less force, hence the usual location of the door handle.
Torque can be either static or dynamic.
A static torque is one which does not produce an angular acceleration. Someone pushing on a closed door is applying a static torque to the door because the door is not rotating about its hinges, despite the force applied. Someone pedaling a bicycle at constant speed is also applying a static torque because they are not accelerating.
The drive shaft in a racing car accelerating from the start line is carrying a dynamic torque because it must be producing an angular acceleration of the wheels given that the car is accelerating along the track.
The magnitude of the torque vector τ for a torque produced by a given force F is
τ=F⋅rsin(θ)
where r is the length of the moment arm and θ is the angle between the force vector and the moment arm. In the case of the door shown in Figure 1, the force is at right angles (90∘ ) to the moment arm, so the sine term becomes 1 and
τ=F⋅r .
The direction of the torque vector is found by convention using the right hand grip rule. If a hand is curled around the axis of rotation with the fingers pointing in the direction of the force, then the torque vector points in the direction of the thumb as shown in Figure 2.
[Explain: Isn't this somewhat arbitrary?]
The SI unit for torque is the Newton-meter.
In imperial units, the Foot-pound is often used. This is confusing because colloquially the pound is sometimes used as a unit of mass and sometimes force. What is meant here is pound-force, the force due to earth gravity on a one-pound object. The magnitude of these units is often similar as 1 Nm≃1.74 ft⋅lbs .
In rotational kinematics, torque takes the place of force in linear kinematics. There is a direct equivalent to Newton’s 2ⁿᵈ law of motion (F=ma ),
τ=Iα .
The concept of rotational equilibrium is an equivalent to Newton’s 1ˢᵗ law for a rotational system. An object which is not rotating remains not rotating unless acted on by an external torque. Similarly, an object rotating at constant angular velocity remains rotating unless acted on by an external torque.
The concept of rotational equilibrium is particularly useful in problems involving multiple torques acting on a rotatable object. In this case it is the net torque which is important. If the net torque on a rotatable object is zero then it will be in rotational equilibrium and not able to acquire angular acceleration.
Exercise 1:
Consider the wheel shown in Figure 3, acted on by two forces. What magnitude of the force F2 will be required for the wheel to be in rotational equilibrium?
There is considerable confusion between torque, power and energy. For example, the torque of an engine is sometimes incorrectly described as its 'turning power'.
Torque and energy have the same dimensions (i.e. they can be written in the same fundamental units), but they are not a measure of the same thing. They differ in that torque is a vector quantity defined only for a rotatable system.
Power however, can be calculated from torque if the rotational speed is known. In fact, the horsepower of an engine is not typically measured directly, but calculated from measured torque and rotational speed. The relationship is:
P=Force⋅DistanceTime=F⋅2πrt=2πτω(ω in revolutions/sec)=τω(ω in radian/sec)
Along with horsepower, the peak torque produced by a vehicle engine is an important and commonly quoted specification. Practically speaking, peak torque is relevant for generally describing how quickly a vehicle will accelerate and its ability to pull a load. Horsepower (relative to weight) on the other hand is more relevant to the maximum speed of a vehicle.
[Explain]
It is often necessary to increase or decrease the torque produced by a motor to suit different applications. Recall that the length of a lever can increase or decrease the force on an object at the expense of the distance through which the lever must be pushed. Similarly, the torque produced by a motor can be increased or decreased through the use of gearing. An increase in torque comes with a proportional decrease in rotational speed. The meshing of two gear teeth can be viewed as equivalent to the interaction of a pair of levers as shown in Figure 5.
The use of adjustable gearing is necessary to obtain good performance in vehicles powered by combustion engines. These engines produce maximum torque only for a narrow range of high rotational speeds. Adjustable gearing allows sufficient torque to be delivered to the wheels at any given rotational speed of the engine.
Bicycles require gearing because of the inability of humans to pedal with a cadence sufficient to achieve a useful speed when driving a wheel directly (unless one is cycling a penny-farthing).
[Explain]
Adjustable gearing is not typically required in vehicles powered by steam engines or electric motors. In both cases, high torque is available at low speeds and is relatively constant over a wide range of speeds.
Exercise 2a:
Cyclist : Hansen, E.A, Smith G. Factors affecting cadence choice during submaximal cycling and cadence influence on performance. International Journal of Sports Physiology and Performance. March 2009; 4(1):3-17.
Diesel engine: Mercedes 250 CDI
Otto cycle engine: Mercedes E250
Electric motor: Tesla Model S 85
Steam locomotive: 2-8-0 "Consolidation" Locomotive at 70% boiler capacity
Penny-farthing : Wikimedia Commons
Sep 12, 2022 · The magnitude of the torque on the disk is rFsin \(\theta\).When \(\theta\) = 0°, the torque is zero and the disk does not rotate. When \(\theta\) = 90°, the torque is maximum and the disk rotates with maximum angular acceleration.
Dec 30, 2017 · Learn the torque formula in three different ways: as the moment of force, as the rate of change of angular momentum, and as the cross product of moment of inertia and angular acceleration. See the derivation of each formula with diagrams and examples.
Apr 6, 2023 · Learn how to calculate torque as the product of force and distance from the axis of rotation. Find out the types, formulas, units, and examples of torque in physics and engineering.
Nov 23, 2023 · Learn the dimensional formula of torque (M L 2 T -2) and how it is derived from moment of inertia and angular acceleration. Find out the dimensions of other related concepts and FAQs on torque.
