Chapter 9: Mechanics
Forces and equilibrium
Newton’s laws and kinematics are developed in Chapter 7. In physics we stress the free-body diagram: draw the body, then each force acting on it (weight , normal reaction , tension , friction , applied forces). For equilibrium, and . A couple is a pair of equal, opposite, parallel forces with moment . The centre of gravity is the point where the weight acts; a body is stable if its centre of gravity lies above its base.
Momentum and energy
Linear momentum is conserved in an isolated system. In an elastic collision kinetic energy is also conserved; in an inelastic collision it is not. Work–energy: . Power: . Efficiency .
Rotational motion
| Linear | Rotational |
|---|---|
| displacement | angle |
| velocity | angular velocity |
| acceleration | angular acceleration |
| mass | moment of inertia |
| force | torque |
| momentum | angular momentum |
| kinetic energy | rotational kinetic energy |
The moment of inertia depends on how mass is distributed: . Common cases: thin ring ; solid disc or cylinder ; solid sphere ; thin rod about its centre . Conservation of angular momentum explains why a spinning skater speeds up when pulling in her arms ( decreases, so increases).
Worked example. A solid disc of mass 2 kg and radius 0.5 m has . A torque of gives .
Circular motion
For uniform circular motion at speed on a circle of radius : , and the centripetal acceleration is , directed towards the centre; the centripetal force is provided by tension, friction, gravity, or normal reaction depending on the situation. A car on a banked road with no friction has .
Gravitation
Newton’s law of gravitation: with . The gravitational field strength is ; at Earth’s surface . The gravitational potential at distance from a mass is .
For a circular orbit, gravity provides the centripetal force: , so , and (Kepler’s third law). The escape speed from a body of radius is .
Example: for Earth, . For a satellite 400 km above the surface, m, so . A geostationary orbit has h, which gives m (an altitude near 36 000 km). The escape speed from Earth’s surface is about .
Common mistakes
Omitting forces from the free-body diagram, or including forces the body exerts on others.
Treating centripetal force as an extra force instead of the net inward force.
Using the Earth’s diameter instead of the radius in , or forgetting to add altitude to the radius.
Practice questions
A 1000 kg car rounds a bend of radius 50 m at . Find the centripetal force. [4500 N]
Find the moment of inertia of a thin ring of mass 0.4 kg and radius 0.3 m. [0.036 kg m²]
A satellite has period 90 minutes. Explain, using , whether its orbit is higher or lower than a geostationary one. [Lower]