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04_Work

1. Work

1.1 Definitions and Properties

  • Mechanical Definition: \(W = F d \cos\theta = \vec{F} \cdot \vec{d}\) (for constant force \(\vec{F}\) and displacement \(\vec{d}\)).

  • Energy Transfer Definition: Energy transferred between an object and a force.

    • Positive work: Energy is transferred to the object.

    • Negative work: Energy is transferred from the object.

  • Unit: Joule (J) = \(N \cdot m\).

Important Notes:

  1. If displacement \(\vec{d} = 0\), then \(W = 0\), even if a force \(\vec{F}\) is applied.

  2. If the force is perpendicular to the displacement (\(\vec{F} \perp \vec{d}\)), then \(W = 0\).

(Mathematical Reminder: The scalar product of two vectors in an orthonormal basis where \(\hat{e}_i \cdot \hat{e}_j = \delta_{ij}\) is given by \(\vec{A} \cdot \vec{B} = \sum_i A_i B_i\).)

1.2 Work Done by a Varying Force

For a force that varies with position, take the limit as \(\Delta x \rightarrow 0\) (\(\Delta W = F_x \Delta x\)). The total work is the integral of the force over the path, representing the area under the \(F_x\) vs. \(x\) curve:

\[ W = \int_{x_i}^{x_f} F_x dx \]

1.3 Examples of Work

Example I: Gravitational Force

Calculating the work done by gravity as an object moves from \(r_i\) to \(r_f\).

Given: \(F = -1.3 \times 10^{22} / r^2\), \(r_i = 1.5 \times 10^{11}\text{ m}\), \(r_f = 2.3 \times 10^{11}\text{ m}\).

\[ W = \int_{r_i}^{r_f} \left(\frac{-1.3 \times 10^{22}}{r^2}\right) dr = 1.3 \times 10^{22} \left[ \frac{1}{r} \right]_{r_i}^{r_f} \]
\[ W = 1.3 \times 10^{22} \left( \frac{1}{2.3 \times 10^{11}} - \frac{1}{1.5 \times 10^{11}} \right) \approx -3 \times 10^{10} \text{ J} \]

Example II: Hooke's Law (Force of Springs)

\[ F_s = -kx \]

Where \(k\) is the force constant (spring constant/stiffness) and \(x\) is the displacement from the relaxed position. The negative sign indicates the force always opposes the displacement.

Measuring \(k\): Hang a mass \(m\) on the spring. At equilibrium, the spring displacement is \(d\).

\[ kd = mg \Rightarrow k = mg/d \]

Work done by a spring:

\[ W = \int_{x_i}^{x_f} (-kx) dx = -\frac{1}{2}k(x_f^2 - x_i^2) \]

  • Case 1: From \(x_i = 0\) to \(x_f = x_{max} \Rightarrow W = -\frac{1}{2}kx_{max}^2\)

  • Case 2: From \(x_i = 0\) to \(x_f = -x_{max} \Rightarrow W = -\frac{1}{2}kx_{max}^2\) (Yields the same sign).

  • Case 3: From \(x_i = -x_{max}\) to \(x_f = x_{max} \Rightarrow W = 0\) (Returns to original state, net work is zero).

Work done by a slow external force:

If an external force stretches a spring slowly enough that the system remains in equilibrium at all times (\(F_{ext} = -F_s\)):

\[ W_{ext} = -W_s \]

2. Kinetic Energy and the Work-Kinetic Energy Theorem

Motivation: Why define Work as \(\vec{F} \cdot \vec{d}\)?

Consider a block of mass \(m\) accelerating constantly from initial velocity \(v_i\) to final velocity \(v_f\) over a distance \(d\) and time \(t\) under a constant force \(F\).

Using kinematics:

\[ v_f = v_i + at \Rightarrow a = F/m \]
\[ d = v_i t + \frac{1}{2}at^2 = \frac{1}{2}(v_i + v_f)t \]

Eliminating \(t\):

\[ v_f = v_i + a \left(\frac{2d}{v_i + v_f}\right) \Rightarrow \frac{1}{2}(v_f^2 - v_i^2) = ad \]

Substituting \(a = F/m\):

\[ \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = Fd \]

The right side is Work (\(W\)). The left side defines the change in a new quantity, Kinetic Energy (\(K\)).

\[ \text{Definition of Kinetic Energy: } K \equiv \frac{1}{2}mv^2 \]

Work-Kinetic Energy Theorem

The net work done on an object equals its change in kinetic energy. This holds true for any force, constant or non-constant:

\[ K_f - K_i = W \quad (\text{for } \forall F) \]

General Proof using Calculus:

\[ W = \int_{x_i}^{x_f} F dx = \int_{x_i}^{x_f} ma dx = \int_{x_i}^{x_f} m \frac{dv}{dt} dx \]

Using the chain rule (\(\frac{dv}{dt} dx = dv \frac{dx}{dt} = v dv\)):

\[ W = \int_{v_i}^{v_f} m v dv = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \]

3. Power

Power is the time rate at which work is done.

  • Average Power: \(P_{avg} = \frac{W}{\Delta t}\)

  • Instantaneous Power: \(P = \frac{dW}{dt}\)

  • For a constant force: \(P = \vec{F} \cdot \frac{d\vec{r}}{dt} = \vec{F} \cdot \vec{v}\)

Units of Power and Energy:

  • Watt (W): \(1\text{ W} = \frac{1\text{ J}}{1\text{ s}} = 1\text{ kg}\cdot\text{m}^2/\text{s}^3\).

  • Kilowatt-hour (kWh): A unit of energy, often used in electricity.

    \[ 1\text{ kWh} = 1\text{ kW} \cdot 1\text{ h} = 10^3\text{ W} \cdot 3600\text{ s} = 3.6 \times 10^6\text{ J} \]