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Session Length
~11 min
Adaptive Checks
10 questions
Transfer Probes
5
Energy, work, and power form a tightly connected triad at the heart of mechanics. Work is defined as the transfer of energy when a force causes a displacement: = F d cos heta$, where $ heta$ is the angle between the force and displacement vectors. Only the component of force along the displacement does work, which is why carrying a box horizontally at constant height involves zero work against gravity even though effort is required to hold it up.
Kinetic energy ( = frac{1}{2}mv^2$) quantifies the energy of motion, while gravitational potential energy ( = mgh$) captures stored energy due to position in a gravitational field. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: {net} = Delta KE$. When only conservative forces act, total mechanical energy is conserved, enabling powerful shortcuts: instead of tracking forces and accelerations, we equate energy at two points. Non-conservative forces like friction convert mechanical energy into thermal energy, reducing the total mechanical energy of the system.
Power measures the rate of energy transfer: = frac{W}{t} = Fv$. A machine that does the same work faster operates at higher power. Understanding energy concepts is essential for analyzing everything from roller coasters and pendulums to collisions and spring systems, and lays the groundwork for thermodynamics and modern physics.
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The transfer of energy when a force acts on an object through a displacement. Calculated as = Fdcos heta$, where $ is the force magnitude, $ is the displacement, and $ heta$ is the angle between them. Work is measured in joules (J).
Example: Pulling a sled 10 m with a rope at 30 degrees above horizontal with 50 N of tension does = 50(10)cos 30^circ = 433$ J of work.
The energy an object possesses due to its motion, given by = frac{1}{2}mv^2$. Since velocity is squared, doubling speed quadruples kinetic energy. Kinetic energy is always positive and is a scalar quantity.
Example: A 1200 kg car traveling at 20 m/s has = frac{1}{2}(1200)(20)^2 = 240{,}000$ J of kinetic energy.
Energy stored due to an objects position in a gravitational field, calculated as = mgh$ near Earths surface, where $ is the height above a chosen reference level. The choice of reference level is arbitrary because only changes in PE matter physically.
Example: A 5 kg book on a 2 m high shelf has = 5(9.8)(2) = 98$ J of gravitational potential energy relative to the floor.
In a system with only conservative forces (gravity, springs), the total mechanical energy = KE + PE$ remains constant. Energy transforms between kinetic and potential forms but the sum never changes.
Example: A pendulum at its highest point has maximum PE and zero KE; at the lowest point all PE has converted to KE, and total energy is the same at both positions.
The net work done on an object equals the change in its kinetic energy: {net} = Delta KE = frac{1}{2}mv_f^2 - frac{1}{2}mv_i^2$. This theorem connects the force-displacement perspective with the energy perspective of motion.
Example: A 2 kg block accelerated from rest to 6 m/s requires net work of = frac{1}{2}(2)(6)^2 - 0 = 36$ J.
The rate at which work is done or energy is transferred: = frac{W}{t}$. For an object moving at constant velocity under a force, = Fv$. Power is measured in watts (W), where 1 W = 1 J/s.
Example: An elevator motor lifting a 1000 kg load at 2 m/s must supply = (1000)(9.8)(2) = 19{,}600$ W of power.
Energy stored in a compressed or stretched spring, given by = frac{1}{2}kx^2$, where $ is the spring constant and $ is the displacement from equilibrium. This energy can be fully recovered as kinetic energy when the spring returns to its natural length.
Example: A spring with = 200$ N/m compressed by 0.3 m stores = frac{1}{2}(200)(0.3)^2 = 9$ J of elastic potential energy.
Forces like friction and air resistance that convert mechanical energy into thermal energy. When non-conservative forces act, total mechanical energy decreases: $Delta E_{mech} = W_{nc}$, where {nc}$ is negative work done by dissipative forces.
Example: A box sliding to a stop on a rough surface converts all its kinetic energy to thermal energy via friction: $ frac{1}{2}mv^2 = mu_k mg d$.
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