Notes : Work Done : Definition, Formula, Types (Positive, Negative & Zero Work) and Units | Class 11 Notes

Notes : Work Done : Definition, Formula, Types (Positive, Negative & Zero Work) and Units | Class 11 Notes

1. Mathematical Definition

In everyday language, "work" means any physical or mental exertion. However, in physics, work has a precise mathematical definition based on force and displacement.

Definition: The work done by a constant force is defined as the product of the component of the force in the direction of the displacement and the magnitude of that displacement.

Vector Dot Product Formula:
$W = (F \cos\theta)d = \mathbf{F} \cdot \mathbf{d}$

Variables Explained:

• W = Work done (It is a scalar quantity, meaning it has magnitude but no direction).
• F = Magnitude of the applied force.
• d = Magnitude of the displacement.
• $\theta$ = The angle between the force vector ($\mathbf{F}$) and the displacement vector ($\mathbf{d}$).

2. The Three Conditions for Zero Work ($W = 0$)

According to the formula, no mechanical work is done if any of the following three conditions are met:

1. Zero Displacement ($d = 0$): When a large force is applied, but the object does not move.
   Exam Examples: Pushing hard against a rigid brick wall, or a weightlifter holding a $150\text{ kg}$ mass steadily on his shoulders for 30 seconds. Your muscles still consume internal energy and you get tired, but mechanically, $W = 0$.
2. Zero Force ($F = 0$): When an object undergoes displacement without any external force acting on it.
   Exam Example: A block sliding across a completely smooth, frictionless horizontal table experiences no horizontal force, yet it undergoes displacement.
3. Perpendicular Force and Displacement ($\theta = 90^\circ$): Because $\cos(90^\circ) = 0$, the entire equation becomes zero.
   Exam Example 1: The vertical gravitational force ($mg$) acting on a block moving horizontally along a smooth table does no work.
   Exam Example 2: The Moon orbiting the Earth in a perfect circular path. The Earth's gravitational force acts radially inward, while the Moon's instantaneous displacement is tangential ($\theta = 90^\circ$).

3. Nature of Work: Positive vs. Negative

Work can be either a positive or a negative scalar depending entirely on the angle $\theta$:

• Positive Work ($0^\circ \le \theta < 90^\circ$): Occurs when the component of the force is in the same direction as the displacement ($\cos\theta$ is positive). The force helps the motion.
• Negative Work ($90^\circ < \theta \le 180^\circ$): Occurs when the component of the force opposes the displacement ($\cos\theta$ is negative).
  Exam Example: Frictional force. Kinetic friction always acts exactly opposite to the direction of displacement ($\theta = 180^\circ$). Since $\cos(180^\circ) = -1$, the work done by friction is always negative.

4. Dimensions and Units

• Dimensional Formula: Work and energy share identical dimensions: $[ML^2T^{-2}]$.
• Standard SI Unit: Joule (J), named after the British physicist James Prescott Joule.

Alternative Units of Work and Energy

Different branches of science use alternative units depending on the scale of the system being studied:

• Erg (erg)
  Value: $10^{-7}\text{ J}$
  Context: This is the absolute unit of work in the CGS (Centimetre-Gram-Second) system, used for small classical mechanics measurements.
• Electron Volt (eV)
  Value: $1.6 \times 10^{-19}\text{ J}$
  Context: Used extensively in atomic and nuclear physics to measure the incredibly tiny energy states of subatomic particles like electrons and protons.
• Calorie (cal)
  Value: $4.186\text{ J}$
  Context: Primarily used in thermodynamics, chemistry, and nutrition. It represents the heat energy required to raise the temperature of 1 gram of water by $1^\circ\text{C}$.
• Kilowatt Hour (kWh)
  Value: $3.6 \times 10^{6}\text{ J}$ (or $3.6\text{ megajoules}$)
  Context: Used in electrical engineering and commercial power. This is the standard "unit" your electricity board uses to measure and bill household power consumption.

5. Frequently Asked Questions (FAQs)

Q1: Why do we get tired when pushing a wall if the work done is zero?

Ans: While the mechanical work on the wall is zero (because displacement $d = 0$), your body is still performing internal biological work. Your muscle fibers are continuously contracting and relaxing, which converts internal chemical energy into thermal energy, making you feel exhausted.

Q2: Is work a vector quantity since it is calculated using force and displacement, which are both vectors?

Ans: No, work is a scalar quantity. It is defined as the scalar dot product of two vectors ($\mathbf{F} \cdot \mathbf{d}$). The dot product of any two vectors always results in a scalar value.

Q3: Can the work done by a centripetal force ever be non-zero?

Ans: No. In a perfect circular motion, centripetal force is always directed toward the center of the path, while the instantaneous displacement is along the tangent. The angle $\theta$ between them is always $90^\circ$. Since $\cos(90^\circ) = 0$, the work done by a centripetal force is strictly zero.

6. Quick Quiz (Multiple Choice Questions)

Q1. A coolie carries a heavy suitcase on his head and walks 50 m along a horizontal railway platform. The work done by the gravitational force on the suitcase is:

• (A) Positive
• (B) Negative
• (C) Zero
• (D) Infinite

Q2. The commercial unit of electricity consumption is the kilowatt-hour (kWh). Its value in standard SI units (Joules) is:

• (A) $3.6 \times 10^{-6}\text{ J}$
• (B) $3.6 \times 10^{6}\text{ J}$
• (C) $1.6 \times 10^{-19}\text{ J}$
• (D) $4.186\text{ J}$

Q3. A ball is thrown vertically upwards into the air. As the ball rises, the signs of the work done by the lifting force during the launch and the work done by gravity during its ascent are, respectively:

• (A) Positive, Positive
• (B) Negative, Positive
• (C) Negative, Negative
• (D) Positive, Negative

Click to view Answer Key & Explanations

1. (C) Zero — The force of gravity acts vertically downwards while the displacement vector is completely horizontal. Since the angle ($\theta$) is $90^\circ$, $$\cos(90^\circ) = 0$.

2. (B) $3.6 \times 10^{6}\text{ J}$ — $1\text{ kWh} = 1000\text{ W} \times 3600\text{ s} = 3,600,000\text{ Joules}$.

3. (D) Positive, Negative — The lifting force acts upwards in the same direction as the launch displacement ($\theta = 0^\circ$, positive). Conversely, during the ascent, gravity pulls downwards while the ball moves upwards ($\theta = 180^\circ$, negative).