AP Physics C Mechanics- 6.2 Torque and Work- Study Notes- New Syllabus
AP Physics C Mechanics- 6.2 Torque and Work – Study Notes
AP Physics C Mechanics- 6.2 Torque and Work – Study Notes – per latest Syllabus.
Key Concepts:
- Energy Transfer by Torque
- Work Done by a Variable Torque
- Work from the Torque–Angle Graph
Energy Transfer by Torque
A torque can transfer energy into or out of an object or rigid system if it is applied over an angular displacement. This is the rotational equivalent of a force doing work when applied over a linear displacement.
When torque acts through an angular displacement, it can increase or decrease the system’s rotational kinetic energy.
Key Idea: – If torque and angular displacement are in the same direction, the system gains energy (positive work). – If they are in opposite directions, the system loses energy (negative work).
\( \mathrm{W = \tau \,\Delta \theta} \quad (\text{for constant torque}) \)
Example
A constant torque of \( \mathrm{4.0\,N·m} \) acts on a wheel causing it to rotate through an angle of \( \mathrm{2.0\,rad} \). Find the work done by the torque on the wheel.
▶️ Answer / Explanation
Step 1: Use the formula for work done by constant torque:
\( \mathrm{W = \tau \,\Delta \theta} \)
Step 2: Substitute values:
\( \mathrm{W = (4.0)(2.0) = 8.0\,J} \)
Result: The torque does \( \mathrm{8.0\,J} \) of positive work on the wheel, increasing its rotational energy.
Work Done by a Variable Torque
If the torque acting on a rigid system is not constant, the total work done is found by integrating torque over the angular displacement through which it acts.
\( \mathrm{W = \displaystyle \int_{\theta_1}^{\theta_2} \tau\,d\theta} \)
- \( \mathrm{W} \): work done by torque (in joules, J)
- \( \mathrm{\tau} \): instantaneous torque (in \( \mathrm{N·m} \))
- \( \mathrm{d\theta} \): infinitesimal angular displacement (in radians)
Key Idea: This relationship is the rotational analog of \( \mathrm{W = \int F\,dx} \) for linear motion. It connects the changing torque to the total energy transferred to or from a rotating system.
Example
A torque varies with angular displacement as \( \mathrm{\tau = 3\theta} \), where \( \mathrm{\tau} \) is in \( \mathrm{N·m} \) and \( \mathrm{\theta} \) is in radians. Find the work done by this torque as the object rotates from \( \mathrm{0} \) to \( \mathrm{2.0\,rad} \).
▶️ Answer / Explanation
Step 1: Use the general work–torque relation:
\( \mathrm{W = \int_{\theta_1}^{\theta_2} \tau\,d\theta} \)
Step 2: Substitute \( \mathrm{\tau = 3\theta} \):
\( \mathrm{W = \int_0^2 3\theta\,d\theta = 3\int_0^2 \theta\,d\theta} \)
Step 3: Integrate:
\( \mathrm{W = 3\left[\tfrac{1}{2}\theta^2\right]_0^2 = 3(2) = 6\,J} \)
Result: The torque performs \( \mathrm{6\,J} \) of work on the system as it rotates through \( \mathrm{2.0\,rad} \).
Work from the Torque–Angle Graph
The work done on a rigid system by a given torque can also be determined from a graph of torque as a function of angular position.
Key Principle: The area under the \( \mathrm{\tau} \) vs. \( \mathrm{\theta} \) curve represents the total work done by the torque:
\( \mathrm{W = \text{Area under the } \tau\text{–}\theta \text{ curve}} \)
- Positive area (above the θ-axis) → positive work (energy gained)
- Negative area (below the θ-axis) → negative work (energy lost)
Example
The torque on a rotating disk increases linearly from \( \mathrm{0\,N·m} \) at \( \mathrm{\theta = 0} \) to \( \mathrm{4.0\,N·m} \) at \( \mathrm{\theta = 2.0\,rad} \), then remains constant until \( \mathrm{\theta = 4.0\,rad} \). Determine the total work done by the torque.
▶️ Answer / Explanation
Step 1: The work equals the area under the \( \mathrm{\tau\text{–}\theta} \) curve.
The graph consists of:
- A triangle from \( \mathrm{0} \) to \( \mathrm{2\,rad} \)
- A rectangle from \( \mathrm{2} \) to \( \mathrm{4\,rad} \)
Step 2: Find each area:
Triangle: \( \mathrm{A_1 = \tfrac{1}{2}(2)(4.0) = 4.0\,J} \)
Rectangle: \( \mathrm{A_2 = (2)(4.0) = 8.0\,J} \)
Step 3: Total work:
\( \mathrm{W = A_1 + A_2 = 4.0 + 8.0 = 12.0\,J} \)
Result: The total work done by the torque is \( \mathrm{12.0\,J} \).