Torque and Angular Acceleration

Torque and angular acceleration: Meaning

The torque can also be expressed in terms of rotational dynamics, where it is needed to produce an angular acceleration on an object causing the object to rotate. Translational torque is the product of perpendicular distance and the tangential force acting on a point. The torque of the rotating object shown in figure 1 can be found using Newton’s second law and linear torque.

We express torque T in terms of the translational force F_t 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.

\[T = F_t \cdot r \cdot \sin{\theta}\]

Where \(\sin{\theta} = 1 \text{ as } \theta = 90^\circ \space F = ma \Rightarrow F_t = ma_t\)

We utilise the translational acceleration a_t 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.

\[a_t = a \cdot r \qquad F_t = m \cdot (a \cdot r)\]

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:

\[T = (mar) \cdot r \cdot \sin{\theta} = m \cdot r^2 \cdot a\]

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.

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.

\[T[Nm] = I[kg \space m^2] \cdot a [rad/s^2]\]

where \(I[kg \space m^2] = m \cdot r^2\)

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.

\[T [Nm] = I \cdot a \qquad T = I \cdot \frac{d\omega}{dt} \qquad L = I \cdot \omega \qquad T = \frac{dL}{dt}\]

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

\[\tau = \sum T = T_1 + T_2 + ... + T_n\]

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.

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.

If the direction of the torque with respect to the point of rotation is clockwise, the torque can be assumed to be negative and vice versa because the angles in an anticlockwise direction are considered positive.