Work, Power, and the Work-Energy Theorem in Physics

Work and Power

Definition of Work

Work is a fundamental concept in physics that quantifies the energy transferred by a force acting over a distance. The infinitesimal work δU performed by a force \( \vec{F} \) during a displacement \( d\vec{r} \) is defined as:

• Formula: $\delta U = \vec{F} \cdot d\vec{r}$

• Work is zero if:

  • No motion: $d\vec{r} = \vec{0}$

  • No force: $\vec{F} = \vec{0}$

  • Force is perpendicular to displacement: $\vec{F} \perp d\vec{r}$

• Work is a scalar quantity and its unit is the Joule (J), where 1 J = 1 N·m.

Types of Forces and Work

• Reactive force: $\delta W = 0$

• Active force: $\delta W \neq 0$

• Driving (actuator) force: $\delta W > 0$

• Resistive (dissipative) force: $\delta W < 0$

Work in Different Coordinate Systems

• Cartesian coordinates:

  • $\vec{F} = F_x \hat{x} + F_y \hat{y} + F_z \hat{z}$

  • $d\vec{r} = dx \hat{x} + dy \hat{y} + dz \hat{z}$

  • $\delta U = F_x dx + F_y dy + F_z dz$

• Cylindrical coordinates:

  • $\vec{F} = F_r \hat{e}_r + F_\theta \hat{e}_\theta + F_z \hat{z}$

  • $d\vec{r} = dr \hat{e}_r + r d\theta \hat{e}_\theta + dz \hat{z}$

  • $\delta U = F_r dr + F_\theta r d\theta + F_z dz$

Total Work

The total work U1→2 performed by a force as a particle moves from state 1 to state 2 is:

• $U_{1 \rightarrow 2} = \int_1^2 \delta U$

Example: Work Associated with Weight

When a body of mass m moves vertically from altitude y1 to y2, the work done by gravity is:

• $\delta U = \vec{W} \cdot d\vec{r} = -mg \hat{y} \cdot d\vec{r} = -mg dy$

• $U_{1 \rightarrow 2} = \int_{y_1}^{y_2} -mg dy = -mg(y_2 - y_1)$

• The horizontal movement does not contribute to this work.

• If the body rises, work is negative; if it falls, work is positive.

• The work depends only on the initial and final positions, not the path—gravity is a conservative force.

Example: Work Associated with a Spring Force

For a spring with force \( \vec{F} = -kx \hat{x} \):

• $\delta U = \vec{F} \cdot d\vec{r} = -kx dx$

• $U_{1 \rightarrow 2} = \int_{x_1}^{x_2} -kx dx = \frac{k}{2}(x_1^2 - x_2^2)$

• The spring force is also a conservative force.

Definition of Power

Power is the rate at which work is done by a force or system:

• $P = \frac{\delta U}{dt}$

• Power is a scalar quantity, measured in Watts (W), where 1 W = 1 J/s.

• Using $\delta U = \vec{F} \cdot d\vec{r}$, power can be expressed as $P = \vec{F} \cdot \vec{v}$.

Work-Energy Theorem

Kinetic Energy

Kinetic energy is the energy a particle possesses due to its motion:

• $K = \frac{1}{2} m v^2 = \frac{1}{2} m \vec{v} \cdot \vec{v}$

• In Cartesian coordinates: $K = \frac{m}{2}(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)$

• In cylindrical coordinates: $K = \frac{m}{2}(\dot{r}^2 + r^2 \dot{\theta}^2 + \dot{z}^2)$

Work-Energy Theorem

The work-energy theorem states that the work done by the resultant force on a particle equals the change in its kinetic energy:

• $\frac{dK}{dt} = P(\vec{R})$

• $\frac{dK}{dt} = \frac{\delta U}{dt} \Rightarrow dK = \delta U$

• Integrating between states 1 and 2: $K_2 - K_1 = U_{1 \rightarrow 2}(\vec{R})$

Applications and Examples

Example: Collar Sliding on a Spiral Rod

A 0.5-kg collar slides with negligible friction along a spiral rod in the vertical plane. The rod has the shape r = 0.3θ (r in meters, θ in radians). The collar is released from rest at A and slides to B under a constant radial force T = 10 N. The velocity v at B can be calculated using the work-energy theorem.

• Holonomic constraint: $r = \alpha \theta$

• Velocity: $\vec{v} = \dot{r} \vec{e}_r + r \dot{\theta} \vec{e}_\theta = \alpha \dot{\theta} \vec{e}_r + \alpha \theta \dot{\theta} \vec{e}_\theta$

• Kinetic energy: $K = \frac{m}{2} \alpha^2 \dot{\theta}^2 (1 + \theta^2)$

• Work done by force T: $U_{A \rightarrow B} = \alpha T (\pi - \frac{\pi}{2}) = \frac{\alpha \pi}{2} T$

• Work-Energy theorem: $K_B - K_A = U_{A \rightarrow B}$

• Since the collar starts from rest, $K_A = 0$.

• Solving for velocity at B: $v_B = \sqrt{\frac{\alpha \pi}{m} T}$

• Numerical values: $\alpha = 0.3$, $m = 0.5$ kg, $T = 10$ N

• Result: $v_B = 4.34$ m/s

Summary Table: Work and Energy Concepts

Concept	Formula	Unit	Notes
Work	$\delta U = \vec{F} \cdot d\vec{r}$	Joule (J)	Scalar, path-dependent for non-conservative forces
Power	$P = \frac{\delta U}{dt} = \vec{F} \cdot \vec{v}$	Watt (W)	Rate of doing work
Kinetic Energy	$K = \frac{1}{2} m v^2$	Joule (J)	Energy due to motion
Work-Energy Theorem	$K_2 - K_1 = U_{1 \rightarrow 2}$	Joule (J)	Work equals change in kinetic energy

Additional info: Academic context and expanded explanations were added to clarify the derivations, coordinate systems, and the physical meaning of work, power, and energy.