Work, Energy & Power
Work, kinetic and potential energy, springs, power, collisions, and conservation laws.
Work Done by a Constant Force
→ DerivationWork is the dot product of force and displacement. Only the component of force along displacement counts.
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Work Done by a Variable Force
→ DerivationWork done when force varies with position — area under the force-displacement graph.
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Work Done by Gravity
→ DerivationWork done by gravity depends only on vertical displacement, not the path taken.
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Work Done by Spring Force
→ DerivationWork done by the spring on the body as it is compressed or stretched by x from natural length.
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Work-Energy Theorem
→ DerivationThe net work done on a body equals its change in kinetic energy.
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Kinetic Energy
→ DerivationEnergy possessed by a body due to its motion.
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Gravitational Potential Energy
→ DerivationEnergy stored in a body due to its position in a gravitational field.
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Elastic Potential Energy
→ DerivationEnergy stored in a spring when compressed or stretched by x from its natural length.
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Hooke's Law
→ DerivationRestoring force of a spring is proportional to displacement from natural length. k is the spring constant.
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Springs in Series
→ DerivationEffective spring constant when springs are connected end to end.
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Springs in Parallel
→ DerivationEffective spring constant when springs share the same extension.
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Spring Constant After Cutting
→ DerivationWhen a spring of constant k and length L is cut to length l, the new constant is k' = kL/l.
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Conservation of Mechanical Energy
→ DerivationTotal mechanical energy is conserved when only conservative forces do work.
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Conservative Force and Potential Energy
→ DerivationForce is the negative gradient of potential energy. Conservative forces derive from a potential.
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Work Done by Non-Conservative Forces
→ DerivationNon-conservative forces change the total mechanical energy of the system.
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Mass-Energy Equivalence
→ DerivationMass and energy are equivalent. A mass m at rest has rest energy mc².
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Average Power
→ DerivationRate of doing work. Energy transferred per unit time.
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Instantaneous Power
→ DerivationPower at a specific instant — dot product of force and velocity.
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Elastic Collision — Final Velocities
→ DerivationVelocities after a perfectly elastic head-on collision. Both momentum and kinetic energy are conserved.
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Perfectly Inelastic Collision
→ DerivationBodies stick together after collision. Maximum kinetic energy is lost.
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Coefficient of Restitution
→ DerivationRatio of relative velocity of separation to relative velocity of approach. e=1 elastic, e=0 perfectly inelastic.
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Kinetic Energy Loss in Perfectly Inelastic Collision
→ DerivationEnergy lost as heat and deformation when two bodies collide and stick together.
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Elastic Collision — Special Cases
→ DerivationEqual masses exchange velocities in elastic collision. Heavy body barely deflected by light one.
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Oblique Elastic Collision
→ DerivationIn 2D elastic collision, momentum is conserved in both x and y directions, and KE is conserved.
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Relation Between KE and Momentum
→ DerivationKinetic energy expressed in terms of momentum. Useful in collision and explosion problems.
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Power of a Vehicle Engine
→ DerivationPower needed to maintain speed v against resistance f_r while accelerating at a.
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