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# Torque

Torque is a turn or twist force acting. Lets have a simply illustration to know that. When we apply the force the door turns on its hinges. Thus a turning effect is produced when we try to open the door. Have you ever tried to do so by applying the force near the hinge?
In the first case, we are able to open the door with ease. In the second case, we have to apply much more force to cause the same turning effect. What is the reason?
This turning effect produced by a force on a rigid body about a point, pivot or fulcrum is called the moment of a force or Torque. Related Calculators Calculate Torque horsepower calculator from torque

## What is Torque?

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Torque is the turning effect of the force about the axis of rotation. The moment of force is called Torque. It is Rotational analogous of Force.

### Torque Definition :

The Rate at which there is a change in angular momentum is also called Torque. It can also be defined as a twist to an object.
When we multiply the magnitude of force or the force's magnitude and the perpendicular distance of the line of action of the force from the axis of rotation then we get a quantity which is called Torque. Mathematically, it can be defined as:
Torque = Perpendicular distance from the axis of rotation x Force.
$\tau$ = r x F.
= r F sin$\theta$.

where,
r = a distance between the point from which torque is measured to the point where force is applied.
$\tau$ is the torque and
F = force applied
Its dimensional formula is [M L2 T -2], which is same as that of work.

### Torque Symbol :

Torque is denoted by the symbol $\tau$. It is a Greek letter. It is pronounced as ‘tau’

## Torque Equation

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Consider a rigid body which is rotating about a given axis of rotation on which force is applied. Let us choose an origin "O" and let "r" be the position vector of the particle on which the force is applied.
Consideration a rigid body, which is rotating about a given axis of rotation AB as shown in figure. Figure b
where
AB is rotational axis,
O is the origin,
O and O' are the two points on the rotational axis,
$\theta$ is the angle between plane AOB and force F

Let F be the force acting on the particle O of the object or the body. F need not necessarily be in the plane ABP. Considering origin O approximately on the axis of rotation.
The torque of F about O is, $\tau$ = $r \times F$ Its component along OA is called the torque of about OA.

To calculate it, we should find the vector $r \times F$ and then find out the angle $\theta$ it makes with OA. The torque about OA is then $|r \times F|$, the torque of a force about a line. This can be shown as given below. Let O be any point on the line AB (Figure b).

The torque of F about O' is,
O' P $\times$ F = (O'O+OP) $\times$ F
= O'O $\times$ F + OP $\times$ F ..................(a)

As this term will have no component along AB. Thus, the component of O'P $\times$ F is equal to that of O'P $\times$ F
Also, $\tau$ = $\frac{dL}{dt}$
which is the rate of change of angular momentum. This torque mentioned here is basically the net torque, i.e.,
$\tau_{1}$ + $\tau_{2}$ + $\tau_{3}$ + $\tau_{4}$ + so on = $\frac{dL}{dt}$
Hence,
$\tau_{net}$ = $\frac{dL}{dt}$ = $\frac{d( I \alpha )}{dt}$
= I $\frac{d \alpha}{dt}$
= I $\alpha$
$\tau$ = I $\alpha$ ........................(c)Where,
I is the moment of inertia and
$\alpha$ is the angular acceleration
This is called Torque Formula

### Relation between Angular Momentum and Torque:

We know that, L= r $\times$ p where,
r = radius, p = momentum L = r $\times$ p ...............(d)

Differentiating the above cross product we get $\frac{dL}{dT}$ = r $\times$ $\frac{dp}{d T}$ + $\frac{dr}{dT}$ $\times$ p
This is due to product rule of differentiation. $\frac{d L}{dT}$ = r x m $\frac{d v}{d T}$ + $\frac{dr}{dT}$ $\times$ m V0) $\frac{dL}{dt}$ = r $\times$ F

$\frac{dL}{dt}$ = $\tau$ ...........................(e)

## Units of Torque

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Torque's dimensional formula is [M L2 T–2], same as that of work.

According to the formula the S.I unit of torque should be Newton meter (N-m).
Other units include :
1. Pound-force-feet (lbf·ft)
2. Foot-pounds-force
3. Inch-pounds-force
4. Ounce-force-inches (oz·in)
5. Meter-kilograms-force: S.I unit
Joule and Torque have same S.I unit which is Nm but it does not mean both both the quantities are same.

## Torque Conversion

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A conversion factor is always necessary.

$Power$ = $Torque \times 2 \pi \times Rotational\ speed$.

Now, let us add units,

$Power(W)$ = $Torque(N-m)\times 2\pi\times Rotational\ speed(rps)$

On dividing the left by 60 seconds per minute and by 1000 watts per kilowatt we get the following:

$Power(W)$ = $\frac{Torque(N.m)\times 2\pi\times Rotational\ speed(rpm)}{60,000}$.

Here, rotational speed is in revolutions per minute (rpm). American automotive engineers sometimes use horsepower for power, foot-pounds (lbf·ft) for torque and rpm for rotational speed. Hence the formula changes to,

$Power(hp)$ = $\frac{Torque(ldf.ft) \times 2\pi\times Rotational\ speed(rpm)}{33,000}$

## Torque vs Horsepower

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The differences between Torque and Horsepower are:
Torque is force multiplied by displacement while Horsepower is the unit of power. Torque is measured in Newton meter (N-m) while Horsepower is itself a unit of energy denoted by hp and is equal to 745.7 k w.

Torque is calculated as
$\tau$ = r x F

While Horsepower is calculated as:
Horse Power = $\frac{\tau \times\ R p m} {5252}$

Torque can be calculated as:
$\tau$ = $\frac{Horse\ power \times 5252}{R p m}$
The above equation gives relationship between Torque and Horsepower.
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