Topic 2: Force and translational dynamics

Dynamics asks not just how objects move but why. This unit links forces to acceleration through Newton's laws, teaches you to draw free-body diagrams, and applies F = ma to friction, inclines, gravitation, and connected systems like the Atwood machine.

Forces and free-body diagrams

A force is a push or pull on an object, measured in newtons (N), where 1 N = 1 kg*m/s^2. Forces are vectors, so they have both magnitude and direction. Common contact forces include the normal force, friction, tension, and applied pushes; the main non-contact force you meet here is gravity. A free-body diagram (FBD) isolates a single object and draws every force acting ON it as an arrow starting from the object, pointing in the direction the force acts. Do not draw forces the object exerts on other things, and do not draw velocity or acceleration on an FBD. Always set up coordinate axes (commonly x horizontal, y vertical, but tilt them along an incline when useful) and resolve angled forces into components. The net force, written as the sum of all forces, determines the object's acceleration.

Newton's first law (inertia)

Newton's first law states that an object at rest stays at rest, and an object in motion continues at constant velocity, unless acted on by a nonzero net force. Inertia is the tendency of an object to resist changes in its motion, and mass is the quantitative measure of inertia: more mass means more resistance to acceleration. A key consequence is that constant velocity (including zero velocity) means the net force is zero, a condition called equilibrium. Equilibrium does not mean no forces act, only that they cancel. For example, a book resting on a table has gravity pulling down and a normal force pushing up that exactly balance, giving zero net force and zero acceleration.

Newton's second law and worked F = ma examples

Newton's second law says the acceleration of an object equals the net force divided by its mass: a = F_net / m, usually written F_net = m*a. Acceleration points in the same direction as the net force. You apply this separately in each direction. Example 1: A 4 kg cart is pushed with a net horizontal force of 12 N. Then a = 12 N / 4 kg = 3 m/s^2. Example 2: A 2 kg box feels a 10 N applied force forward and a 4 N friction force backward. Net force = 10 N - 4 N = 6 N, so a = 6 N / 2 kg = 3 m/s^2 forward. Example 3: To find force, rearrange. A 1500 kg car accelerating at 2 m/s^2 needs F = m*a = 1500 kg * 2 m/s^2 = 3000 N. Always sum forces with correct signs before dividing by mass.

Newton's third law (action-reaction)

Newton's third law states that if object A exerts a force on object B, then object B exerts an equal-magnitude, opposite-direction force on object A. These two forces form an action-reaction pair. The critical rule is that the two forces in a pair act on DIFFERENT objects, so they never cancel each other on a single FBD. When you push on a wall, the wall pushes back on you with equal force. A common trap: the normal force and weight on a book are NOT a third-law pair, because they act on the same object and are not the same type of force; they merely happen to be equal when the surface is horizontal and there is no vertical acceleration.

Weight and normal force

Weight is the gravitational force on an object, given by F_g = m*g, where g is the gravitational field strength (about 9.8 m/s^2 near Earth's surface, often rounded to 10). Mass stays constant everywhere, but weight changes with g, so an astronaut has the same mass on the Moon but weighs less. The normal force, written F_N, is the perpendicular support force a surface exerts on an object pressing into it. On a flat horizontal surface with no vertical acceleration, F_N = m*g. But the normal force is not always equal to weight: in an elevator accelerating upward, F_N = m*g + m*a, so you feel heavier (apparent weight increases); accelerating downward gives F_N = m*g - m*a, and in free fall F_N = 0, producing apparent weightlessness.

Tension

Tension is the pulling force transmitted through a rope, string, cable, or chain. We usually idealize ropes as massless and inextensible, which means the tension has the same magnitude everywhere along the rope and the rope cannot push, only pull. An ideal pulley simply changes the direction of the tension without changing its magnitude. For a hanging object in equilibrium, the tension equals the weight: a 5 kg lamp hanging from one cable has T = m*g = 5 kg * 10 m/s^2 = 50 N. When the system accelerates, tension differs from weight; for a mass pulled upward at acceleration a, the rope must supply T = m*g + m*a.

Static and kinetic friction

Friction opposes relative sliding between surfaces in contact. Static friction acts on objects that are not yet sliding and adjusts itself to prevent motion, up to a maximum value f_s,max = mu_s * F_N, where mu_s is the coefficient of static friction. Once the applied force exceeds this maximum, the object breaks free and kinetic friction takes over with a constant magnitude f_k = mu_k * F_N, where mu_k is the coefficient of kinetic friction. Usually mu_k is less than mu_s, which is why it is harder to start something moving than to keep it moving. Friction depends on the normal force and the surfaces, not on contact area or speed. Example: a 10 kg crate on a floor with mu_k = 0.3 has f_k = 0.3 * (10 kg * 10 m/s^2) = 30 N opposing its motion.

Inclined planes

On a ramp tilted at angle theta, the smart move is to tilt your axes so x runs along the incline and y runs perpendicular to it. Then weight splits into two components: the component along the incline (pulling the object down the slope) is m*g*sin(theta), and the component perpendicular to the incline (pressing into the surface) is m*g*cos(theta). The normal force balances the perpendicular component, so F_N = m*g*cos(theta) on a frictionless incline with no perpendicular acceleration. For a frictionless incline, the acceleration down the slope is a = g*sin(theta), independent of mass. Example: a block on a frictionless 30 degree incline accelerates at a = 10 m/s^2 * sin(30) = 10 * 0.5 = 5 m/s^2. With friction, subtract f_k = mu_k * m*g*cos(theta) from the gravity component before dividing by mass.

Newton's law of gravitation and gravitational field strength

Newton's law of universal gravitation says every two masses attract each other with a force F_g = G*m1*m2 / r^2, where G is the gravitational constant (about 6.67e-11 N*m^2/kg^2) and r is the distance between their centers. The force weakens with the square of distance, so doubling r cuts the force to one quarter. The gravitational field strength g at a distance r from a body of mass M is g = G*M / r^2, measured in N/kg (equivalent to m/s^2). Near Earth's surface this gives about 9.8 m/s^2. The weight you measure, m*g, is just this gravitational field strength multiplied by your mass, which is why g is both 'field strength' and the free-fall acceleration.

Connected systems and the Atwood machine

When objects are linked by ropes, they share the same magnitude of acceleration because the rope is inextensible. A powerful shortcut is to treat the whole connected system as one object: a_system = F_net,external / m_total. Then return to a single object to find internal tension. The Atwood machine is two masses m1 and m2 (with m1 greater) hanging over a frictionless pulley. The net driving force is the weight difference (m1 - m2)*g, and the total mass is m1 + m2, so a = (m1 - m2)*g / (m1 + m2). Example: m1 = 6 kg and m2 = 4 kg with g = 10 m/s^2 gives a = (6 - 4)*10 / (6 + 4) = 20 / 10 = 2 m/s^2. To find tension, analyze the lighter mass: T - m2*g = m2*a, so T = m2*(g + a) = 4*(10 + 2) = 48 N.