6. Torque

Definitions

Torque (τ)

• A measure of rotational force applied to an object that causes it to rotate around an axis, measured in N⋅m

$$\vec{\tau} = \vec{r} \times \vec{F}$$

where $\vec{r}$ is the position vector from the axis to the point of force application, and $\vec{F}$ is the applied force

Rotational Equation of Motion

• The rotational analogue of Newton's second law

$$\vec{\tau} = I \alpha$$

 

Derivations

consider a force ($\vec{F}$) acting on a particle (mass $m$, distance from axis $r$) that causes it to accelerate around an axis

$$\vec{F} = ma_t$$

since the tangential acceleration is related to angular acceleration by

$$a_t = \alpha r$$

substituting this expression

$$\vec{F} = m r \alpha$$

multiply both sides by $r$

$$\vec{F}r = m r^2 \alpha$$

recognizing that $\vec{F}r = \vec{\tau}$ and $mr^2 = I$

$$\vec{\tau} = I \alpha$$

 

Useful Equations

$$\vec{\tau} = \vec{r} \times \vec{F}, \quad \vec{\tau} = I \alpha$$

 

Example 1

A force of 10 N is applied perpendicular to a wrench at a distance of 0.3 m from the bolt. What is the torque applied to the bolt?

Answer

Using $\tau = rF$ (for perpendicular force)

$$\tau = 0.3 \times 10 = 3 \text{ N⋅m}$$

 

Example 2

A wheel with moment of inertia 2 kg⋅m² experiences a torque of 8 N⋅m. What is its angular acceleration?

Answer

Using $\tau = I\alpha$

$$\alpha = \frac{\tau}{I} = \frac{8}{2} = 4 \text{ rad/s}^2$$