Premed · Premed · Physics 1
Lecture 7: Work and Kinetic Energy
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define work done by a constant force and by a variable force
- Calculate work using the dot product and from force-displacement graphs
- State and apply the work-energy theorem
- Calculate kinetic energy and relate changes in kinetic energy to net work
- Compute power as the rate of doing work
Lecture Content
I. Definition of Work
Work (W) is the energy transferred to or from an object by a force acting over a displacement. For a constant force acting over a straight-line displacement, work is calculated as W = F . d = F d cos(theta), where theta is the angle between the force vector and the displacement vector. When the force is parallel to the displacement (theta = 0), the work equals Fd, the maximum positive work. When the force is perpendicular (theta = 90 degrees), the work is zero. When the force opposes the displacement (theta = 180 degrees), the work is -Fd, representing negative work.
Work is a scalar quantity measured in Joules (J), where 1 J = 1 N m = 1 kg m^2/s^2. Work can be positive (energy transferred to the object), negative (energy transferred from the object), or zero.
II. Work by Specific Forces
The work done by different forces follows directly from the definition. Gravity does work W_gravity = -mg Delta y: negative when the object moves up (gravity opposes the displacement) and positive when it moves down (gravity aids the displacement). The normal force on a flat surface is perpendicular to the displacement, so W_N = 0. Tension in a string does work that depends on the angle between the tension and the displacement. Kinetic friction always opposes motion, so its work is always negative: W_friction = -f_k d. This negative work represents the conversion of kinetic energy into thermal energy.
III. Work Done by a Variable Force
When force varies with position, work must be calculated as an integral: W = integral from x_i to x_f of F(x) dx. Graphically, this is the area under the F(x) vs. x curve, where area above the x-axis contributes positive work and area below contributes negative work.
An important example is the spring force described by Hooke's law: F_spring = -kx, where k is the spring constant (in N/m) and x is the displacement from the natural length. This is a restoring force that opposes displacement from equilibrium. The work done by a spring as it stretches or compresses from x_i to x_f is W_spring = (1/2)k x_i^2 - (1/2)k x_f^2.
<image>Panel A: A force vs. displacement graph for a constant force, showing a rectangle whose area equals W = Fd. Panel B: A force vs. displacement graph for a spring (F = -kx), showing a triangle. The shaded area under the curve from 0 to x represents the work done W = (1/2)kx^2. Panel C: A general variable force curve with irregular shape; the shaded area under the curve between x_i and x_f is labeled as the work done by the variable force.</image>
IV. Kinetic Energy
Kinetic energy (K) is the energy an object possesses due to its motion, given by K = (1/2) m v^2. It is a scalar quantity that is always non-negative, measured in Joules. Kinetic energy depends on speed rather than the direction of velocity. Doubling the speed quadruples the kinetic energy, a fact with important implications for vehicle safety. An object at rest has zero kinetic energy.
V. The Work-Energy Theorem
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy: W_net = Delta K = K_f - K_i = (1/2)m v_f^2 - (1/2)m v_i^2. This holds regardless of the nature of the forces involved. The net work W_net is the sum of work done by all forces: W_net = W_gravity + W_normal + W_friction + W_applied + ... If W_net > 0, the object speeds up. If W_net < 0, the object slows down. If W_net = 0, the speed remains constant.
The work-energy theorem provides an alternative to Newton's second law that is especially useful when you know forces and displacements but not time, since it bypasses the need to find acceleration explicitly.
<image>A block sliding down a rough incline from height h. Forces shown: gravity (doing positive work mgh), normal force (doing zero work, perpendicular to motion), and kinetic friction (doing negative work -f_k d). At the top, the block has speed v_i; at the bottom, speed v_f. The work-energy theorem equation is written: mgh - f_k d = (1/2)mv_f^2 - (1/2)mv_i^2. Each work contribution is labeled with its sign and source.</image>
VI. Power
Power is the rate at which work is done or energy is transferred: P = dW/dt. For a constant force acting on an object moving at constant velocity, power is P = F . v = F v cos(theta). Average power is defined as P_avg = W / Delta t.
The SI unit of power is the Watt (W), where 1 W = 1 J/s = 1 kg m^2/s^3. Other common units include horsepower (1 hp = 746 W) and the kilowatt-hour (1 kWh = 3.6 x 10^6 J), which is a unit of energy rather than power. A car engine's power limits its maximum speed: at terminal velocity, the engine power equals the rate of energy dissipation by drag and friction, giving P = F_drag v. Power is also crucial in biological contexts, where metabolic rate and muscle power output govern physical performance.
VII. Problem-Solving with Work and Energy
The strategy for work-energy problems is to identify the system and all forces acting on it, calculate the work done by each force (using W = Fd cos theta or integration), sum all contributions to get W_net, and then apply the work-energy theorem W_net = Delta K to solve for the unknown, which is often the final speed.
This approach offers several advantages over the force/acceleration method. There is no need to find acceleration explicitly, the method works directly with displacements rather than time, and because the work-energy theorem is a scalar equation, there are no vector components to manage. This method forms the foundation for the more general energy conservation approach developed in the next lecture.
<image>A comparison diagram showing two methods to solve the same problem (block sliding down a frictionless incline). Left side: Newton's second law approach requiring FBD, component resolution, finding acceleration, then using kinematics. Right side: Work-energy theorem approach directly computing work by gravity and setting it equal to the change in kinetic energy. Both arrive at the same answer, but the energy method requires fewer steps. Key equations for each step are shown.</image>
