Dynamics: Newton's Laws and Forces
Dynamics: Newton's Laws and Forces
Dynamics explains motion through force. This section tests Newton's three laws, F = ma applied to single particles and to connected bodies, weight and normal reaction, limiting friction, motion on inclined planes, lift problems and the transmission of tension through light inextensible strings.
Newton's laws and force diagrams
First law: a body stays at rest or in uniform motion unless a resultant force acts. Second law: the resultant force vector equals mass times the acceleration vector in an inertial frame. Third law: action and reaction are equal, opposite and act on different bodies. Draw the forces first: weight mg, normal reaction perpendicular to the surface, tension along the string away from the body, friction opposing motion, applied force.
Friction and inclines
In the ideal Coulomb model static friction satisfies |F_s| ≤ μ_s R, with equality at impending slip. During sliding, the kinetic-friction model is |F_k| = μ_k R and its direction opposes relative sliding. On a smooth incline of angle θ, the component of weight along the slope is mg sin θ and perpendicular to it mg cos θ.
Connected particles
Treat the system together to find acceleration, then apply F = ma to one body alone to find tension. For two masses m₁ > m₂ over a smooth pulley, a = (m₁ − m₂)g/(m₁ + m₂). In a lift accelerating upward at a, the reaction satisfies R − mg = ma, so R = m(g + a); apparent weight falls when a is downward.
Worked example
Masses of 6 kg and 4 kg hang from a light inextensible string over a smooth pulley. Take g = 10 m s⁻². Find the acceleration and the tension. For the whole system the driving force is (6 − 4)g = 20 N and the total mass is 10 kg, so a = 2 m s⁻². For the 4 kg mass moving up: T − 4g = 4 × 2, hence T = 40 + 8 = 48 N. Checking the 6 kg mass moving down: 6g − T = 60 − 48 = 12 = 6 × 2, consistent.
Traps that cost marks
- Adding the masses in the numerator when computing the acceleration.
- Using tension equal to the weight of the hanging mass.
- Omitting the normal reaction or resolving weight along the wrong axis on an incline.
- Using μR in a situation that is not limiting.