Work Done by a Force

Definition of Work

• In physics, work is the transfer of energy from one object to another, or the transformation of energy from one form to another.
• A force does work on an object when it acts on that object and causes a displacement in the direction of the force.
• Work is a scalar quantity; it has magnitude but no direction.

KEY TAKEAWAY: Work represents the energy transferred when a force causes displacement.

Calculating Work Done by a Constant Force

• When a constant force acts on an object, the work done is given by:

  \[W = Fs \cos{\theta}\]

  Where:
  * \(W\) is the work done (in Joules, J)
  * \(F\) is the magnitude of the force (in Newtons, N)
  * \(s\) is the magnitude of the displacement (in meters, m)
  * \(\theta\) is the angle between the force vector and the displacement vector

• If the force and displacement are in the same direction (\(\theta = 0^\circ\)), then \(\cos{\theta} = 1\) and the equation simplifies to:

  \[W = Fs\]

• If the force and displacement are perpendicular (\(\theta = 90^\circ\)), then \(\cos{\theta} = 0\) and the work done is zero.

  • Example: A satellite orbiting Earth at a constant speed experiences a gravitational force towards the Earth, but its displacement is tangential to the orbit. The gravitational force does no work on the satellite.

• Negative Work: If the force and displacement are in opposite directions (\(\theta = 180^\circ\)), then \(\cos{\theta} = -1\) and the work done is negative.

  • Example: Friction often does negative work, as it opposes the motion of an object.

EXAM TIP: Always consider the angle between the force and displacement vectors. A common mistake is to assume they are always in the same direction.

Work Done by a Variable Force

• When the force is not constant, we cannot use \(W = Fs\) directly. Instead, we can determine the work done by finding the area under the force vs. displacement graph.

• This method is valid for one-dimensional motion only.

• Area Calculation:

  • For simple shapes (rectangles, triangles), use geometric formulas to calculate the area.
  • For irregular shapes, approximate the area by dividing it into smaller, simpler shapes, or use integration (if applicable).

• Spring Force: A common example of a variable force is the force exerted by a spring, which obeys Hooke’s Law:

  \[F = kx\]

  Where:
  * \(F\) is the force exerted by the spring (in N)
  * \(k\) is the spring constant (in N/m)
  * \(x\) is the displacement from the equilibrium position (in m)

  The work done to stretch or compress a spring is equal to the elastic potential energy stored in the spring:

  \[E_s = \frac{1}{2}kx^2\]

  This is also the area under the force vs. displacement graph for the spring.

STUDY HINT: Practice sketching force vs. displacement graphs for different scenarios and calculating the work done from the area under the graph.

Units of Work

• The SI unit of work is the Joule (J).
• 1 Joule is equal to 1 Newton-meter (N·m).
• Since work is a measure of energy transfer, it has the same units as energy.

Examples

• Lifting an object: A weightlifter lifts a barbell of mass \(m\) to a height \(h\). The work done by the weightlifter is \(W = Fs = mgh\), where \(g\) is the acceleration due to gravity.

• Pushing a box: A person pushes a box across a floor with a force \(F\) over a distance \(s\). If there is friction, the person must do work to overcome the frictional force. The net work done on the box is the sum of the work done by the applied force and the work done by friction.

• Compressing a spring: Compressing a spring a distance \(x\) from its equilibrium position requires work. The work done is stored as elastic potential energy in the spring.

COMMON MISTAKE: Forgetting to convert units to SI units (meters, kilograms, seconds) before performing calculations.

Table: Key Differences

VCAA FOCUS: VCAA exams frequently include questions involving work done by friction, gravity, and spring forces. Pay close attention to these scenarios.