Gravitational potential energy and kinetic energy
Section: Work, Energy and Power | Syllabus: Cambridge AS Level Physics 9702
Gravitational Potential Energy (E_P) Work must be done on an object to lift it to a greater height in a gravitational field. This work is stored as GPE. Δ E_P = mg Δ h Applies when Δ h is small compared to Earth's radius (so g is constant).
Experimental Skills 5.1: Efficiency of an Electric Motor Investigating the efficiency of a motor lifting a mass. Figure 5.11: Motor Experiment Setup Show a motor connected to a power supply and joule meter.
The motor shaft winds a string which lifts a mass hanger. Label 'motor', 'joule meter', 'mass', 'height Δ h'. Method Input Energy (E_in): Measured by Joule meter. Output Energy (E_out): Work done lifting mass (Δ E_P = mg Δ h).
Efficiency = fraction × 100. Kinetic Energy (E_K) Derivation considering an object of mass m accelerated from rest (u=0) by force F. E_K = fractionmv^2 Derivation: Displacement s = fraction(u+v)t = fractionvt (since u=0).
Acceleration a = fraction = fraction. Force F = ma = m(fraction). Work W = Fs = (fraction) × (fraction) = fractionmv^2. Transferring between GPE and KE For a falling object with negligible air resistance: Loss of GPE = Gain of KE mg Δ h = fractionmv^2 With Friction/Air Resistance If there is resistance, some GPE is converted to thermal energy (Work done against friction).
mg Δ h = fractionmv^2 + Work Done against Friction Worked Example: Roller Coaster (Figure 5.13) Question: A roller coaster car (3200 kg) drops from a height of 45 m. Track length is 53 m. Speed at bottom is 24 m s^-1.
Calculations (a) Change in GPE: Δ E_P = 3200 × 9.81 × 45 = 1.4 MJ. (b) Change in KE: Start (2.0 m/s): E_K1 = 0.5 × 3200 × 2.0^2 = 6.4 kJ. End (24 m/s): E_K2 = 0.5 × 3200 × 24^2 = 920 kJ. Gain in KE = 920 - 6.4 910 kJ.
(c) Work Done against Friction: W_fric = Δ E_P - Gain in E_K = 1400 kJ - 910 kJ = 490 kJ. (d) Average Friction Force: F = fraction = fraction = 9.2 kN Examiner Tips Path Independence: For GPE changes, only the vertical height Δ h matters, not the path taken.
Friction path: For work done against friction, the actual distance travelled along the path (s) must be used, not the vertical height.
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