Work by Constant Force — Revision Notes

⚡ 30-Second Revision

Definition: — $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$
Units: — Joule (J) = Newton-meter (N$\cdot$m)
Scalar Quantity: — Work has magnitude only, no direction.
Positive Work: — $0^\circ \le \theta < 90^\circ$ (Force component in direction of displacement)
Negative Work: — $90^\circ < \theta \le 180^\circ$ (Force component opposite to displacement)
Zero Work: — $\theta = 90^\circ$ (Force perpendicular to displacement) or $d=0$.
Common Zero Work Forces: — Normal force, centripetal force, tension (if perpendicular to displacement).
Work by Gravity: — $-mgd$ (upward motion), $+mgd$ (downward motion).
Work by Friction: — Always negative, $W_f = -f_k d$.

2-Minute Revision

Work done by a constant force is a measure of energy transfer, defined as the scalar product of the force vector and the displacement vector: $W = Fd \cos\theta$. Here, $F$ is the force magnitude, $d$ is the displacement magnitude, and $\theta$ is the angle between their directions.

Work is a scalar quantity, measured in Joules (J). It can be positive (force aids motion, $\theta < 90^\circ$), negative (force opposes motion, $\theta > 90^\circ$), or zero (force is perpendicular to motion, $\theta = 90^\circ$, or no displacement).

Key scenarios for zero work include the normal force on a horizontal surface, centripetal force in uniform circular motion, and any force acting without causing displacement. For problems involving vector components, use $W = F_x d_x + F_y d_y + F_z d_z$.

Remember that friction always does negative work. This concept is fundamental for understanding the Work-Energy Theorem and energy conservation.

5-Minute Revision

Work done by a constant force is a crucial concept in physics, representing the energy transferred to or from an object by a force. The defining equation is $W = \vec{F} \cdot \vec{d}$, which expands to $W = Fd \cos\theta$.

Here, $F$ is the magnitude of the constant force, $d$ is the magnitude of the displacement, and $\theta$ is the angle between the force and displacement vectors. Work is a scalar quantity, meaning it only has magnitude, and its SI unit is the Joule (J), where $1,\text{J} = 1,\text{N} \cdot \text{m}$.

Understanding the sign of work is vital:

1
Positive Work: — Occurs when the force has a component in the direction of displacement ($0^\circ \le \theta < 90^\circ$). This means the force adds energy to the object. Example: Pushing a box forward.
2
Negative Work: — Occurs when the force has a component opposite to the direction of displacement ($90^\circ < \theta \le 180^\circ$). This means the force removes energy from the object. Example: Work done by friction or air resistance.
3
Zero Work: — Occurs when the force is perpendicular to the displacement ($\theta = 90^\circ$) or when there is no displacement ($d=0$). Example: Work done by the normal force on a horizontally moving object, or by centripetal force in uniform circular motion.

When forces and displacements are given in vector form, the work done is calculated using the dot product of their components: $W = F_x d_x + F_y d_y + F_z d_z$. This method is often simpler than finding magnitudes and angles separately.

Worked Example: A $4,\text{kg}$ block is pulled $3,\text{m}$ horizontally by a $20,\text{N}$ force at an angle of $60^\circ$ above the horizontal. Calculate the work done by the applied force.

Given: $F = 20,\text{N}$, $d = 3,\text{m}$, $\theta = 60^\circ$.
Formula: $W = Fd \cos\theta$
Calculation: $W = (20,\text{N})(3,\text{m})\cos(60^\circ) = 60 \times 0.5 = 30,\text{J}$.

This positive work indicates energy is transferred to the block by the pulling force. For NEET, always draw free-body diagrams to correctly identify forces and angles, and be meticulous with signs and units.

Prelims Revision Notes

Work by Constant Force: NEET Revision Notes

1. Definition and Formula:

- Work ($W$) is the scalar product of constant force ($\vec{F}$) and displacement ($\vec{d}$). - $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ - $F$: magnitude of force, $d$: magnitude of displacement, $\theta$: angle between $\vec{F}$ and $\vec{d}$. - If $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ and $\vec{d} = d_x \hat{i} + d_y \hat{j} + d_z \hat{k}$, then $W = F_x d_x + F_y d_y + F_z d_z$.

2. Units and Dimensions:

- SI Unit: Joule (J). $1,\text{J} = 1,\text{N} \cdot \text{m}$. - CGS Unit: Erg. $1,\text{J} = 10^7,\text{erg}$. - Dimensional Formula: $[M L^2 T^{-2}]$. (Same as energy).

3. Types of Work:

- Positive Work: $0^\circ \le \theta < 90^\circ$. Force component aids motion. Energy added. (e.g., pulling a block). - Negative Work: $90^\circ < \theta \le 180^\circ$. Force component opposes motion. Energy removed. (e.g., friction, air resistance, gravity when lifting). - Zero Work: $\theta = 90^\circ$ or $d=0$. No energy transfer by that force. (e.g., normal force, centripetal force, carrying an object horizontally).

4. Key Scenarios & Examples:

- Work by Gravity: - Lifting object (upward displacement): $W_g = -mgd$ (as $\theta = 180^\circ$). - Falling object (downward displacement): $W_g = +mgd$ (as $\theta = 0^\circ$). - Horizontal motion: $W_g = 0$ (as $\theta = 90^\circ$). - Work by Friction: Always negative, $W_f = -f_k d$ (as $\theta = 180^\circ$). - Work by Normal Force: Always zero for motion along the surface ($W_N = 0$). - Work by Centripetal Force: Always zero for uniform circular motion ($W_c = 0$).

5. Work-Energy Theorem (for constant forces):

- Net work done by all forces equals the change in kinetic energy: $W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$.

6. Problem-Solving Tips:

- Draw FBDs to correctly identify forces and angles. - Be careful with vector components and dot product calculations. - Pay attention to the sign of work; it indicates energy flow. - Ensure all units are consistent (SI preferred).

Vyyuha Quick Recall

Work Is For Doing Calculations On Signs: $W = Fd \cos\theta$. (W, I, F, D, C, O, S for Work, Is, Force, Displacement, Cosine, Theta, Sign)