Work, Kinetic Energy, and Conservation of Energy

Definition and Calculation of Work

• What is Work in Physics? In everyday language, "work" is any effort you make. In physics, work specifically means using a push or pull (a force) to move an object through a distance. If you push hard against a solid wall but it does not move, you have done zero physics work!

• Work Done by a Constant Force: Calculated using the dot product between the force and displacement:

$W = \mathbf{F} \cdot \Delta \mathbf{s} = F\Delta s\cos(\theta)$

• Understanding the Angle ($\theta$): Only the component of the force pointing in the direction of movement does work. The angle $\theta$ is the angle between the push and the direction of movement:

  • Pushing along motion ($\theta = 0^\circ$ to $\theta < 90^\circ$): Work is positive ($W > 0$). Energy enters the object, speeding it up.

  • Pushing sideways ($\theta = 90^\circ$): Work is zero ($W = 0$). No energy is added or removed, so speed stays constant (for example, holding a heavy box while walking horizontally).

  • Pushing against motion ($90^\circ < \theta \le 180^\circ$): Work is negative ($W < 0$). Energy is taken away from the object, slowing it down (for example, friction or braking).

• Work Done by a Variable Force: When force changes during the motion, integrate over the distance:

$W = \int_{s_i}^{s_f} \mathbf{F} \cdot d\mathbf{s}$

• Units: Measured in Joules ($\text{J}$) or Newton-meters ($\text{N}\cdot\text{m}$). Energy conversion: $1\,\text{calorie} = 4.2\,\text{J}$.

Kinetic Energy and the Work-Kinetic Energy Theorem

• Kinetic Energy ($K$): The energy an object possesses strictly because it is moving. Any object with mass $m$ moving at speed $v$ has kinetic energy:

$K = \frac{1}{2}mv^2$

• Key Takeaway on Speed: Because speed is squared ($v^2$), doubling your speed quadruples your energy ($4\times$), and tripling your speed gives nine times the energy ($9\times$).

• Work-Kinetic Energy Theorem: The net work done on an object by all combined forces equals its change in kinetic energy:

$W_{\text{net}} = \Delta K = \frac{1}{2}m(v_f^2 - v_i^2)$

• Positive net work ($W_{\text{net}} > 0$) increases speed.

• Negative net work ($W_{\text{net}} < 0$) decreases speed.

• Zero net work ($W_{\text{net}} = 0$) keeps speed constant.

Potential Energy and Conservative Forces

• Potential Energy ($U$): "Stored" energy waiting to be released.

• Conservative Forces: Forces where the energy stored depends only on the starting and ending positions, not the path taken (for example, gravity and ideal springs). Friction is non-conservative because a longer path wastes more energy as heat.

• Potential Energy Relation: Stored potential energy change equals the negative work done by that conservative force:

$\Delta U = -W_c = -\mathbf{F} \cdot \Delta \mathbf{s}$

• Gravitational Potential Energy ($U_G$): Stored when lifting an object against gravity to height $h$:

$U_G = mgh$

• Path Independence of Gravity: Dropping an object vertically, sliding it down a frictionless ramp, or swinging it on a pendulum from height $h$ results in the identical final speed at the bottom:

$v = \sqrt{2gh}$

Spring Force and Elastic Potential Energy

• Hooke's Law (Spring Force): A stretched or compressed spring pulls or pushes back toward its relaxed equilibrium point:

$F_s = -kx$

• The negative sign indicates the spring force always opposes displacement.

• The spring constant $k$ measures spring stiffness (stiffer springs have higher $k$).

  • Elastic Potential Energy ($U_s$): Energy stored in the spring when stretched or compressed by distance $x$:

$U_s = \frac{1}{2}kx^2$

• Compressing or stretching a spring twice as far ($2x$) stores four times the potential energy ($4U_s$).

Conservation of Energy

• The Big Idea: Energy cannot be created or destroyed; it only changes forms (like converting stored height energy into speed, or speed into heat via friction).

• Master Energy Equation:

$\Delta K + \Delta U + f\Delta s = W_{\text{ext}}$

• Non-conservative friction work: $W_{\text{nc}} = -f\Delta s$ (converts mechanical energy to heat).

• Conservative force work: $W_c = -\Delta U$.

  • Isolated System ($W_{\text{ext}} = 0$): When no outside pushes occur, total mechanical energy is conserved:

$E_i = E_f$

$\Delta K + \Delta U + f\Delta s = 0$

Questions & Discussion

• Oblique Angle Work: A force of $10\,\text{N}$ over $5\,\text{m}$ at an angle of $127^\circ$ does negative work:

$W = 10\,\text{N} \times 5\,\text{m} \times \cos(127^\circ) = -30\,\text{J}$

• Opposing Angle Work: A force of $10\,\text{N}$ over $5\,\text{m}$ directly opposing motion at $180^\circ$:

$W = 10\,\text{N} \times 5\,\text{m} \times \cos(180^\circ) = -50\,\text{J}$

• Signs of Work for Elevators:

  • Gravity on an upward-moving elevator: Negative (gravity pulls down while elevator goes up).

  • Cable tension on a downward-moving elevator: Negative (cable pulls up while elevator goes down).

  • Cable tension on a stationary elevator: Zero (displacement is zero).

  • Gravity on an elevator falling freely: Positive (gravity and motion are in the same direction).

• Speed Scaling: Tripling speed increases kinetic energy by a factor of $3^2 = 9$, yielding $9K_i$.

• High-Speed Vehicle Braking (Granny Problem):

  • Granny goes $50\,\text{mph}$ and barely stops before an obstacle. Another car goes $70\,\text{mph}$ and brakes at the same point.

  • Because energy scales with speed squared, the $70\,\text{mph}$ car has roughly double the energy: $70^2 / 50^2 \approx 4900 / 2500 \approx 2$.

  • Braking dissipates the first $2500$ energy units (stopping Granny), leaving the second car with $2400$ energy units: $\sqrt{2400} \approx 49\,\text{mph} \approx 50\,\text{mph}$ at collision.

• Stopping Distance on a Rough Surface: A block descending frictionless height $h$ onto a flat rough surface stops after distance:

$\Delta s = \frac{h}{\mu_k}$

• Bead on a Wire: Starting from rest at point A and dropping vertical height $0.8\,\text{m}$ gives:

$v = \sqrt{2gh} = \sqrt{2(9.8\,\text{m/s}^2)(0.8\,\text{m})} \approx 4\,\text{m/s}$

Formula Sheet for Beginners

• Work (Constant Force):

$W = F\Delta s\cos(\theta)$

• Work (Variable Force):

$W = \int_{s_i}^{s_f} \mathbf{F} \cdot d\mathbf{s}$

• Kinetic Energy:

$K = \frac{1}{2}mv^2$

• Work-Kinetic Energy Theorem:

$W_{\text{net}} = \Delta K = \frac{1}{2}m(v_f^2 - v_i^2)$

• Gravitational Potential Energy:

$U_G = mgh$

• Spring Force (Hooke's Law):

$F_s = -kx$

• Spring Elastic Potential Energy:

$U_s = \frac{1}{2}kx^2$

• Free-Fall Speed from Rest:

$v = \sqrt{2gh}$

• Thermal Energy from Friction:

$E_{\text{th}} = f\Delta s = \mu_k mg\Delta s$

• Total Conservation of Energy:

$K_i + U_i + W_{\text{ext}} = K_f + U_f + f\Delta s$

Variable Guide and When to Use Each Formula

1. Work Formula: $W = F\Delta s\cos(\theta)$

   • Letters Explained:

   • $W$ = Work done on the object, in Joules ($\text{J}$).

   • $F$ = Magnitude of the applied force, in Newtons ($\text{N}$).

   • $\Delta s$ = Distance or displacement the object moves, in meters ($\text{m}$).

   • $\theta$ = Angle between the force direction and the movement direction, in degrees.

   • When to Use: Use whenever a constant push or pull acts on an object over a straight distance.

2. Kinetic Energy Formula: $K = \frac{1}{2}mv^2$

   • Letters Explained:

   • $K$ = Kinetic energy of motion, in Joules ($\text{J}$).

   • $m$ = Mass of the object, in kilograms ($\text{kg}$).

   • $v$ = Speed of the object, in meters per second ($\text{m/s}$).

   • When to Use: Use whenever you need to calculate how much motion energy an object has at a given speed, or to find speed from known energy.

3. Work-Kinetic Energy Theorem: $W_{\text{net}} = \Delta K$

   • Letters Explained:

   • $W_{\text{net}}$ = Total net work from all combined forces, in Joules ($\text{J}$).

   • $\Delta K$ = Change in kinetic energy ($K_f - K_i$), in Joules ($\text{J}$).

   • When to Use: Use when forces act on an object and you want to connect those forces directly to how much the object speeds up or slows down.

4. Gravitational Potential Energy: $U_G = mgh$

   • Letters Explained:

   • $U_G$ = Gravitational potential energy, in Joules ($\text{J}$).

   • $m$ = Mass of the object, in kilograms ($\text{kg}$).

   • $g$ = Acceleration due to gravity ($9.8\,\text{m/s}^2$ on Earth).

   • $h$ = Vertical height above reference ground level, in meters ($\text{m}$).

   • When to Use: Use whenever an object changes vertical elevation (height).

5. Hooke's Law: $F_s = -kx$

   • Letters Explained:

   • $F_s$ = Restoring force exerted by the spring, in Newtons ($\text{N}$).

   • $k$ = Spring constant (stiffness of the spring), in Newtons per meter ($\text{N/m}$).

   • $x$ = Distance the spring is stretched or compressed from equilibrium, in meters ($\text{m}$).

   • When to Use: Use when you need to find the instantaneous force exerted by a spring at a specific stretch or compression.

6. Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$

   • Letters Explained:

   • $U_s$ = Stored elastic potential energy, in Joules ($\text{J}$).

   • $k$ = Spring constant, in Newtons per meter ($\text{N/m}$).

   • $x$ = Distance stretched or compressed, in meters ($\text{m}$).

   • When to Use: Use whenever a spring is compressed or stretched and you need to know how much stored energy is inside it.

7. Free-Fall Speed: $v = \sqrt{2gh}$

   • Letters Explained:

   • $v$ = Final speed at the bottom, in meters per second ($\text{m/s}$).

   • $g$ = Gravitational acceleration ($9.8\,\text{m/s}^2$).

   • $h$ = Vertical height dropped, in meters ($\text{m}$).

   • When to Use: Use as a quick shortcut when an object starts from rest and converts all height into speed without friction.

8. Conservation of Energy Master Equation: $K_i + U_i + W_{\text{ext}} = K_f + U_f + f\Delta s$

   • Letters Explained:

   • $K_i, K_f$ = Initial and final kinetic energies ($\frac{1}{2}mv^2$).

   • $U_i, U_f$ = Initial and final potential energies (gravity $mgh$ and/or spring $\frac{1}{2}kx^2$).