Honors Physics Unit 4 Review: Forces and Newton’s Laws of Motion

A force is a push or pull that one object exerts on another. It's a vector, so it has a size and a direction, and its SI unit is the newton (N), where $1 \text{ N} = 1 \text{ kg·m/s}^2$.

Contact forces need touching. These include the normal force (a surface pushing perpendicular to itself), tension (a rope pulling along its length), friction (parallel to the surface, opposing sliding or attempted sliding), and applied pushes or pulls. Field forces act at a distance. Gravity, electric, and magnetic forces are field forces, and near Earth the gravitational force on an object is its weight, $W = mg$ with $g = 9.8 \text{ m/s}^2$.

To analyze any situation, draw a free-body diagram. Pick one object, draw it as a dot or box, and add an arrow for every force acting on it. Only include forces on that object, not forces it exerts on other things.

Worked example. A crate is pulled right with a 30 N rope while friction pulls left with 12 N. Gravity (100 N down) and the normal force (100 N up) cancel vertically. Horizontally, take right as positive: $F_{net} = 30 - 12 = 18 \text{ N}$ to the right. The crate accelerates to the right.

Newton's first law says an object at rest stays at rest, and an object in motion keeps the same speed and direction, unless a net external force acts on it. The tendency to resist changes in motion is inertia, and mass is the measure of it. A bowling ball is harder to start or stop than a tennis ball because it has more mass.

The law covers two cases with one rule. Zero net force means zero acceleration, so a parked car and a car cruising at a steady 25 m/s on a straight road are both in equilibrium.

If moving objects keep moving, why does a rolling ball stop? Rolling resistance and air resistance can produce a net force that slows it. A ball can roll without sliding, so sliding friction is not always the explanation. Static friction keeps surfaces from starting to slide and can grow up to a maximum of $\mu_s N$. Kinetic friction acts once sliding starts and equals $\mu_k N$.

Worked example. You push a 20 kg box across a floor where $\mu_k = 0.30$. On a level floor with only vertical forces of gravity and the normal force, $N = mg = 20 \times 9.8 = 196 \text{ N}$. Kinetic friction is $f_k = 0.30 \times 196 = 58.8 \text{ N}$. To keep the box sliding at constant velocity, you push with exactly 58.8 N, so the net force is zero.

Newton's second law gives the numbers: $F_{net} = ma$, or $a = F_{net}/m$. Acceleration is directly proportional to net force and inversely proportional to mass. Double the net force and acceleration doubles, and double the mass and acceleration halves. Acceleration always points in the direction of the net force, not necessarily the direction of motion. A car braking while moving forward has a backward net force and a backward acceleration.

This law also separates mass from weight. Mass (kg) is the amount of matter and stays fixed. Weight (N) is the gravitational force, $W = mg$. A 10 kg object weighs 98 N on Earth and about 17 N on the Moon, where $g \approx 1.7 \text{ m/s}^2$.

The usual method is to draw a free-body diagram, choose a positive direction, add forces along each axis, and then apply $F_{net} = ma$ to each axis.

Worked example. A 4.0 kg cart is pulled with 20 N to the right while friction pushes 6.0 N to the left. First, $F_{net} = 20 - 6.0 = 14 \text{ N}$ to the right. Then, $a = \frac{14}{4.0} = 3.5 \text{ m/s}^2$ to the right. Using 20 N alone would give 5.0 m/s², which is wrong because it ignores friction.

Newton's third law says that if object A exerts a force on object B, then B exerts a force on A that is equal in size and opposite in direction: $F_{A \text{ on } B} = -F_{B \text{ on } A}$. The two forces in a pair act at the same time, are the same type of force, and act on different objects.

Walking works because your foot pushes back on the ground and the ground pushes you forward, which requires friction. A rocket pushes exhaust gas backward, and the gas pushes the rocket forward, so it doesn't need air to push against.

A book resting on a table shows the classic trap. The book's weight (Earth pulls book down) and the normal force (table pushes book up) are equal, but they are not a third-law pair because both act on the book. The partner of the book's weight is the book pulling Earth up. The partner of the normal force is the book pushing down on the table.

Worked example. A 1000 kg car and a 4000 kg truck collide, and each exerts 20,000 N on the other. If other forces are negligible, the car's acceleration is $\frac{20{,}000}{1000} = 20 \text{ m/s}^2$, and the truck's is $\frac{20{,}000}{4000} = 5.0 \text{ m/s}^2$ in the opposite direction. The equal collision forces produce different accelerations because the vehicles have different masses.