# Torque and Angular Acceleration

If you're interested in how things move, then you might have heard of torque and angular acceleration. These two things are really important in translational dynamics, which is all about how objects move in a straight line.

So what is torque? Well, imagine you're trying to turn a wheel. You need to apply a force to make it spin, right? Torque is the measure of how much force you need to apply to make an object rotate around a turning point. It's measured in Newton-metres, which is a fancy way of saying how hard you need to push on something and how far away from the turning point you are.

The principle of moments is also really important in translational dynamics. It basically says that when an object is not moving, the sum of clockwise moments around a turning point is the same as the sum of anticlockwise moments around that point. This helps us understand how objects stay in place and don't fall over.

But what about angular acceleration? Well, that's all about how quickly an object is rotating. It's like regular acceleration, but for things that are spinning. The faster something is spinning, the higher its angular acceleration.

So, there you have it! Torque and angular acceleration are two really important concepts in translational dynamics. By understanding how they work together, we can learn a lot about how things move and why they don't fall over.

## Torque and angular acceleration: Meaning

If you're interested in how things rotate, then you might have heard of torque. It's a really important concept in rotational dynamics, which is all about how objects spin.

So, what is torque in rotational dynamics? Well, it's the measure of how much force you need to apply to make an object rotate. It's like a kind of push that makes something spin around an axis. The more torque you apply, the faster the object will rotate.

Translational torque is a bit different. It's the product of the distance from a point and the force that's acting on it. This kind of torque is needed to make an object move in a straight line.

In order to find the torque of a rotating object, we can use Newton's second law and linear torque. This helps us figure out how much force we need to apply to make something rotate around an axis.

So, there you have it! Torque is really important in rotational dynamics because it helps us understand how things spin. By figuring out how much torque we need, we can make objects rotate faster or slower. And by understanding translational torque, we can make objects move in a straight line.

We express torque T in terms of the translational force Ft in Newtons and the radius r in metres. We then apply Newton’s second law. In the object shown above, we get an expression for translational force in terms of mass and acceleration, where m is the mass and a the acceleration in m/s2.

T= Ft ·r· sinθ  Where sinθ=1 as θ=90० F=ma ⇒Ft=mat

We utilise the translational acceleration at in relation to the derived equation, which expresses the translational acceleration in terms of the radius, and we substitute it into the equation of the translational force.

a t  = a · r  F t  = m · ( a · r   )

We then use the torque formula noted above but substitute the derived translational force expression. For an angle of 90 degrees, sinθ is equal to one. The torque is, therefore, equal to the product of mass, acceleration, and squared radius, as shown below:

T = ( m a r ) · r · sin θ = m ·  r 2  · a

The expression that was derived can be written in terms of the moment of inertia I measured in kgm2 multiplied by acceleration measured in m/s2, as the moment of inertia is the mass of an object multiplied by the squared distance to the axis of rotation.

T = ( m ·  r 2  ) · a

T =   I · a

## What is the torque formula for rotational dynamics?

As it was derived above, the torque for rotational motion is defined as the product of the moment of inertia and angular acceleration, as shown below, where T is the torque measured in Newton-metres. I is the moment of inertia, which is the rotating object’s tendency to resist angular acceleration. Angular acceleration α is the rate of change of angular velocity.

T [ N m ] =   I [ k g  m 2  ] · a [ r a d /  s 2  ]

w h e r e   I   [ k g  m 2  ] = m ·  r 2

It can be derived from the formula that the torque depends on the moment of inertia and hence on the distribution of the mass of the object and its distance to the centre of rotation. The angular acceleration is proportional to the magnitude of torque.

The moment of inertia is the rotational equivalent of force about an axis in translational dynamics, and the angular displacement is the equivalent of the linear displacement in rotational dynamics.

Torque is also equal to the rate of change of angular momentum if the mass is conserved. This is derived below, where ω is the angular velocity in rad/s and L the angular momentum. We use the previous formula of torque in terms of the moment of inertia and substitute the acceleration with the rate of change of angular velocity. That gives us the rate of change of angular momentum, which is equal to the torque, as shown below.

T [ N m ] =   I · a     T = I ·   d ω   d t         L =   I · ω       T =   d L   d t

If two or more forces act on an object, the total torque is the vector sum of the torques, as shown below, where n number of torques are present.

∑  T = T 1 + T 2 + . . . +  T n

## Direction of torque

As torque is a vector quantity, both magnitude and direction are needed to define it. The direction of torque can be found using the right-hand rule, where four fingers of the right hand are pointed in the direction of the force F that is applied. The direction of the torque is the same as the direction of the thumb.

Figure 2. Direction of torque when the force is upwards. Source: Georgia Panagi, Study Smarter.

An example is shown in figure 2, where the force applied is upwards, and the resulting torque is shown below. If the force applied is downwards, the resulting torque is in the opposite direction, as shown in figure 3.

Figure 3. The direction of torque when the force is downwards. Source: Georgia Panagi, Study Smarter.

A wheel rotates about axis A. A tangential force of 30 N is applied at the edge of the wheel, which has a radius of 40 cm. The wheel accelerates linearly from rest, reaching a speed of rotation of 10 rad/sec in 8 sec. Determine the moment of inertia of the wheel.

We begin using the torque equation.

T = F · r = 30   N · 0 . 4   m = 12   N m         T = I · a   ⇒ I =   12 / a

To determine the angular acceleration in rad/s2, we need to find the rate of change in angular velocity.

a =   d ω   d t   =   10 - 0   8 s - 0 s   = 1 . 25   r a d /  s 2

We then substitute it into the previous equation to find the moment of inertia:

I =   12   N m   1 . 25   r a d /  s 2 = 9 . 6   k g  m 2

A wheel is shown below in figure 4. Determine the net torque on the wheel about its centre and the angular acceleration of the wheel if the moment of inertia is 20 kgm2. R1 is 5cm, R2 is 12cm, F1 is 15 N (yellow in the figure below), and F2 is 18 N (blue in the figure). We create a diagram to visualise the problem.

The total torque is the vector sum of the individual torques acting on the object. We analyse each torque individually.   T 1 =   F 1 · d · sin θ   =   15   N · 0   m · sin ( 20 ) = 0   N m     T 2 = F 2 · R 2 =   18   N · 0 . 12   m = 2 . 16   N m   ( c l o c k w i s e )   We can see that the total torque created by the forces is equal to the torque created by F1 since the torque created by F2 is zero. We continue by using the torque and angular acceleration relation. Note that the result will be negative as the moment is clockwise. T = I · a ⇒ a =  T I  =   2 . 16   N m   20   k g  m 2    = - 0 . 108   r a d /  s 2

Torque and Angular Acceleration - Key takeaways Torque for rotational dynamics is the product of the moment of inertia and angular acceleration. Moment of inertia is the body’s tendency to resist angular rotation. The magnitude of the moment of inertia is calculated by the sum of the products of the masses that make up a body with the square of their distance from the axis of rotation. Angular displacement is the equivalent of linear displacement in rotational dynamics. The magnitude of the net torque is found by the algebraic sum of the torques acting on the object.

## Torque and Angular Acceleration

What is the relationship between torque and angular acceleration?

Torque is directly proportional to angular acceleration when the rotational inertia is constant.

How do you find angular acceleration from torque?

We can find angular acceleration from torque by dividing torque by the moment of inertia.

Does torque affect angular acceleration?

Yes, torque does affect the angular acceleration.

How do you calculate angular acceleration?

We can calculate angular acceleration by using the torque and moment of inertia relation, which can be mathematically expressed as a = T/I.

What is torque in simple terms?

It is the rotational or twisting force required to rotate an object about a point.

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