Torque and Work

Learning Goals

After this lesson, you should be able to:

• identify when a torque does positive, negative, or zero work;
• calculate work by a constant torque using \(W=\tau\Delta\theta\);
• derive the rotational-work expression from linear work;
• add work contributions from several torques;
• apply the rotational work–energy theorem;
• find work from signed area on a torque–angular-position graph;
• distinguish torque, work, and energy despite related units;
• include thermal transfer or other energy pathways in a system model; and
• design an experiment relating torque, angular displacement, and energy change.

1. A Torque Transfers Energy Through Rotation

For a constant torque exerted through an angular displacement,

The torque and angular displacement must be measured about the same axis and assigned signs using the same clockwise/counterclockwise convention. If the system does not rotate, that torque does no mechanical work during the interval even if it produces stress or balances another torque.

3. Deriving \(W=\tau\Delta\theta\)

Suppose a tangential force \(F_t\) acts at radius \(r\) while its point of application moves through arc length \(s\). Linear work is

Using \(s=r\Delta\theta\) and \(\tau=rF_t\),

The angle must be in radians because \(s=r\Delta\theta\) is valid in this form only for radian measure. A radial force has \(F_t=0\), so it produces neither torque nor rotational work about that axis.

4. The Sign of Rotational Work

Positive work does not mean counterclockwise. A clockwise torque doing work during clockwise displacement also gives a positive product because both quantities have the same sign.

5. Worked Example: Constant Torque

A constant \(4.5\,\mathrm{N\,m}\) torque acts in the direction of rotation while a flywheel turns through \(3.2\,\mathrm{rad}\).

Answer: The torque does \(+14.4\,\mathrm J\) of work and transfers that amount of energy into the modeled flywheel. If the torque opposed the same displacement, the work would be \(-14.4\,\mathrm J\).

6. Large Torque Does Not Guarantee Large Work

Work depends on both torque and angular displacement. Compare:

Case B has the smaller torque but greater work because the torque acts through a larger angle. A torque applied to a locked shaft can be large while its work remains zero.

7. Several Torques: Add Signed Work

A motor supplies \(+6.0\,\mathrm{N\,m}\), a brake supplies \(-2.0\,\mathrm{N\,m}\), and axle friction supplies \(-0.50\,\mathrm{N\,m}\) while a wheel rotates through \(+5.0\,\mathrm{rad}\). Each torque is constant.

Equivalently, \(\tau_{\mathrm{net}}=+3.5\,\mathrm{N\,m}\) and \(W_{\mathrm{net}}=\tau_{\mathrm{net}}\Delta\theta=17.5\,\mathrm J\).

8. Rotational Work–Energy Theorem

The net work done by all external torques changes the rotational kinetic energy of a fixed-axis rigid system:

This equation relates two states without requiring the elapsed time. It is especially useful when torque and angular displacement are known but angular acceleration or time is not.

9. Worked Example: Find the Final Angular Speed

A flywheel with \(I=0.80\,\mathrm{kg\,m^2}\) initially rotates at \(3.0\,\mathrm{rad/s}\). The net work done on it is \(+16\,\mathrm J\).

Answer: The final angular-speed magnitude is \(7.0\,\mathrm{rad/s}\). Energy alone does not specify clockwise or counterclockwise direction; that must come from the motion history or a stated sign.

10. Torque–Angular-Position Graphs

On a graph of torque \(\tau\) versus angular position \(\theta\), work is the signed area between the curve and the horizontal axis.

• Area above the axis is positive work for increasing \(\theta\).
• Area below the axis is negative work for increasing \(\theta\).
• Equal positive and negative areas give zero net work.
• The torque value at one point is not enough; the entire interval matters.

The units of graph area are \((\mathrm{N\,m})(\mathrm{rad})=\mathrm J\).

11. Worked Graph Example Without Calculus

From \(\theta=0\) to \(2.0\,\mathrm{rad}\), torque increases linearly from \(2.0\) to \(8.0\,\mathrm{N\,m}\). The area is a rectangle plus a triangle:

If the torque then remains at \(-3.0\,\mathrm{N\,m}\) from \(2.0\) to \(5.0\,\mathrm{rad}\), that interval contributes

The net work from \(0\) to \(5.0\,\mathrm{rad}\) is therefore \(+1.0\,\mathrm J\).

12. Torque and Energy Share \(\mathrm{N\,m}\), but Are Different

Both torque magnitude and work can be expressed using \(\mathrm{N\,m}\), but they describe different ideas:

Torque can exist at one instant without any energy transfer; work requires displacement through an interval.

13. Axis and System Choices Must Stay Consistent

1. Choose the rigid system and rotation axis.
2. Draw all external forces at their application points.
3. Calculate every torque about the same axis.
4. Choose a positive rotational sense.
5. Use angular displacement about that same axis in radians.
6. Decide whether frictional or applied interactions cross the system boundary.
7. Add signed work contributions or use net torque when valid.
8. Check whether the answer represents work by one torque, net work, or a total energy change.

14. Friction and Thermal Energy

A friction torque often does negative work on the rotating rigid system, decreasing its mechanical energy. If a wheel with \(K_i=25\,\mathrm J\) receives \(+8.0\,\mathrm J\) of motor work while friction transfers \(5.0\,\mathrm J\) to thermal energy, then

Whether thermal energy is described as remaining inside a larger system or leaving a smaller mechanical system depends on the chosen boundary. State that boundary before claiming mechanical energy is conserved.

15. Connection to the Linear Work Model

The analogy helps organize equations, but torque is not force and angular displacement is not linear distance.

16. Investigation: Work from a Torque–Angle Graph

Use a low-friction wheel or turntable, string wrapped around a known axle radius, force sensor or spring scale, angle markings or rotary sensor, balance, and movable masses.

1. Choose the rotating system, axis, and positive angular direction.
2. Pull the string slowly through a measured angle while recording tangential force and angular position.
3. Convert force to applied torque with \(\tau=rF_t\).
4. Plot torque against angular position and estimate work from rectangular or trapezoidal areas.
5. Repeat with an opposing friction pad or elastic element so torque varies with angle.
6. Measure initial and final angular speeds in a dynamic trial and compare graph area with \(\Delta K_{\mathrm{rot}}\).
7. Discuss axle friction, string stretch or slip, changing effective radius, sensor zero offset, and angular-resolution uncertainty.

Do not use only average force unless the data justify how that average represents the full torque–angle area.

17. Common Traps and Key Takeaways

• Work requires angular displacement; torque alone is not work.
• Use radians in \(W=\tau\Delta\theta\).
• Match the torque axis and angular-displacement axis.
• Positive work means torque and displacement share a rotational sense, not necessarily counterclockwise.
• Use net work for \(\Delta K_{\mathrm{rot}}\).
• Include negative areas on torque–angle graphs.
• Do not confuse the slope of a torque–angle graph with work; work is area.
• Do not double-count friction as both negative mechanical work and an additional unexplained energy loss.
• Energy gives angular-speed magnitude, not rotation direction by itself.