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 hApplies 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 SetupShow 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 = mg Δ hE_in × 100.
Kinetic Energy (E_K)
Derivation considering an object of mass m accelerated from rest (u=0) by force F.
E_K = (1/2)mv^2Derivation:
- Displacement s = (1/2)(u+v)t = (1/2)vt (since u=0).
- Acceleration a = (v-u/t) = (v/t).
- Force F = ma = m((v/t)).
- Work W = Fs = ((mv/t)) × ((vt/2)) = (1/2)mv^2.
Transferring between GPE and KE
For a falling object with negligible air resistance:
Loss of GPE = Gain of KEmg Δ h = (1/2)mv^2
With Friction/Air Resistance
If there is resistance, some GPE is converted to thermal energy (Work done against friction).
mg Δ h = (1/2)mv^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 = (W/s) = (490,000/53) = 9.2 kNExaminer 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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