Understanding Work in Physics

Work

• Work is useful in physics for analyzing scenarios with force and displacement.
• Work equals the component of force in the direction of displacement.
• When force and displacement aren't in the same direction, work = force × displacement × cosine(θ).
• Work is measured in Joules (J).

Example calculation

• A force of 20 N at a 30° angle displaces a box 3m, resulting in Work = 20 N * 3 m * cos(30°) ≈ 52 J.

Sign of work

• Work is positive when force and displacement are in the same direction.
• Work is zero when there is force but no displacement, or force is perpendicular to displacement.
• Work is negative when force opposes displacement (e.g., friction).

Example: Positive work

• Pulling a box with 20 N in the direction of its 3 m displacement results in Work done = 20 N * 3 m = 60 J.

Example: Zero Work

• Holding a box up involves force but no displacement, so the work done is zero.
• Carrying a box sideways also yields zero work, as force is upwards, while displacement is sideways.

Example: Negative Work

• Friction of 10 N acts on a box displaced 1.5 m to the right (positive direction), so Work done = -10 N * 1.5 m = -15 J.

Graphical Representation of Work

• Work can be found on a force vs. position graph by calculating the area under the curve.
• For constant force, the area is a rectangle (base x height); for variable force, the area can be divided into shapes like triangles and rectangles.

Example: Constant Force

• A constant force of 3 N applied from position 2 m to 4 m results in Work = 3 N * (4 m - 2 m) = 6 J.

Example: Non-Constant Force

• When force increases from 3 N to 5 N over a displacement from 2 m to 4 m, Area is a triangle (0.5 * 2 N * 2 m) + rectangle (3 N * 2 m) = 2 J + 6 J = 8 J.

Net Work

• The sum of the work done by all forces acting on a system.
• Net work = net force × displacement × cos(θ).
• With an applied force of 20 N, friction of 10 N, and displacement of 3 m, Net force = 20 N - 10 N = 10 N, and Net work = 10 N * 3 m = 30 J.

Work-Energy Theorem

• Derivation starts with work = net force × displacement
• Utilizing Newton's 2nd Law (F=ma) and a kinematic equation (vf² = vi² + 2ad)
• Net work = final kinetic energy - initial kinetic energy

Kinetic Energy

• Kinetic energy (KE) = 1/2 * mass * velocity².

Work Energy Theorem

• The net work done on a system equals the change in the system's kinetic energy.

Applying the Work-Energy Theorem

• With a box displaced 3 m and a net work of 30 J, starting from rest with a mass of 10 kg, then 30 J = 1/2 * 10 kg * vf², solving for the final velocity: vf ≈ 2.4 m/s.

Work against Gravity

• Lifting an object requires work against gravity.
• Work Equation: Force equals the weight of the object (mg) times the height change (h).
• Gravitational potential energy (GPE) = mass * g * height.
• Only the change in GPE is significant.
• Path independence: GPE change depends only on height change, not the path taken.

Conservative Forces

• A force where work depends only on the start and end points of the motion, such as gravity and spring force.
• Work done by a conservative force leads to a decrease in potential energy.
• -ΔPE = ΔKE.

Conservation of Mechanical Energy

• When only conservative forces are present, total mechanical energy (potential + kinetic) remains constant.
• KEinitial + PEinitial = KEfinal + PEfinal

Non-Conservative Forces

• Forces where the work done depends on the path taken (e.g., friction).
• Non-conservative forces add or remove mechanical energy from a system.
• Energy can dissipate as thermal energy.

System Comparison

• In a system without non-conservative forces, energy transfers completely between forms. (Pendulum Example)
• In a system with non-conservative forces, energy is lost to the environment.

Updated Energy Equation

• Network = Work done by non-conservative forces + Work done by conservative forces
• Using the Theorem: Wnc + Wc = ΔKE
• Since WC equals -ΔPE: Wnc = ΔKE + ΔPE
• Written out: KEinitial + PEinitial + Wnc = KEfinal + PEfinal

Work Done by Non-Conservative Forces

• Can be positive, negative, or zero.
• Positive increases mechanical energy.
• Negative decreases mechanical energy
• Zero means mechanical energy is conserved

Examples

• Positive : Pushing a box with applied force overcomes friction, increasing kinetic energy.
• Negative : Friction slows a sliding box, decreasing kinetic energy.
• Zero : Pushing a box at a constant velocity leads to mechanical energy being conserved

Power

• Power: Rate at which work is done (or energy is transferred).
• Formula: Power = Work / Time = Change in energy / Time.
• Elevator Example: Lifting a 2000 kg elevator 28 m in 42 seconds requires Power = (m * g * displacement) / time ≈ 1.3 * 10^4 Watts.
• Alternative calculation: Power equals Force times Velocity when they're calculated in the same direction.
• Unit: Watts (W).
• Box Example: Pulling an object with 12 N moving at 3 m/s uses Power equals 12 N * 3 m/s = 36 W.

Explore the concept of work in physics, defined as the force component in the direction of displacement. Learn to calculate work using the formula involving force, displacement, and the angle between them. Understand positive, negative, and zero work scenarios with practical examples.