Newton's Laws and Application of Newton's Laws

Vectors and scalars

A physical quantity is anything in physics that can be measured, such as time, mass, weight, force and charge. Some quantities are fully described by a number and a unit; others also need a direction.

A mass of 5 kg is complete as it stands, but a force of 5 N is not: it matters whether you push the box to the left or to the right. Weight is a force, so it is a vector that always points downwards, while mass is a scalar.

Writing vectors. A symbol with an arrow above it, F→, represents the force vector: its magnitude and its direction. The same letter without the arrow, F, represents only the magnitude of the force, for example F = 20 N.

Drawing vectors. A vector is drawn as an arrow. The length of the arrow, drawn to a chosen scale, shows the magnitude, and the arrowhead shows the direction. The start of the arrow is its tail and the end with the arrowhead is its head.

Two force vectors drawn to scale: 40 N to the right and 20 N to the left. The first arrow is twice as long as the second because its force is twice as large.

Directions in one dimension. Along a straight line there are only two directions. Choose one as positive (for example to the right or east); the opposite direction is then negative. A force of −20 N, with right as positive, is 20 N to the left.

Adding vectors in one dimension

Equal vectors have the same magnitude and the same direction. The negative of a vector has the same magnitude but the opposite direction: if A is 30 N to the right, −A is 30 N to the left.

A and B are equal vectors (30 N to the right). C is the negative of A (30 N to the left). D points the same way as A but is larger, so it is not equal to A.

The resultant of force vectors is also called the net force. It can be found by drawing or by calculation.

Forces of 15 N east, 40 N west and 10 N east drawn tail to head, with east to the right. F₂ points the way with the larger total, so it goes on the top line. On the line underneath, F₁ and F₃ follow tail to head, starting at the head of F₂. R runs along the bottom line from the start line to the head of F₃. R is 3 squares long, which is 15 N west.

Forces in the same direction add up to a larger resultant; forces in opposite directions partly or completely cancel. For two forces along a line, the largest possible resultant is their sum and the smallest is their difference.

Subtracting vectors. To subtract a vector, add its negative: A − B = A + (−B). This works for up to four force vectors in the same way, one at a time.

Three horizontal forces on a crate. With east as positive, R = (+30 N) + (+12 N) + (−50 N) = −8 N, so the resultant is 8 N west.

The resultant of perpendicular vectors

In Grade 10 all the vectors acted along one straight line. In two dimensions they can act in any direction in a flat plane, so we work on a Cartesian plane: a horizontal x-axis and a vertical y-axis that cross at right angles. East (or to the right) is usually the positive x-direction and north (or upwards) the positive y-direction.

The method works for force vectors and for displacement vectors, and for up to four vectors at a time.

Four forces act on one point. Horizontally: Rₓ = (+20 N) + (−8 N) = +12 N, so Rₓ = 12 N east. Vertically: Rᵧ = (+14 N) + (−5 N) = +9 N, so Rᵧ = 9 N north.

Tail to head: Rᵧ starts at the head of Rₓ, and R closes the right-angled triangle. θ is the angle between R and the x-axis.

Tail to tail: Rₓ and Rᵧ start at the same point. The rectangle is completed with dashed lines, and R is the diagonal from the common tail.

Finding the resultant by drawing. Choose a scale, draw the vectors tail to head with a ruler and protractor, and draw R from the tail of the first vector to the head of the last. Measure the length of R and convert it with the scale; measure θ with the protractor. This graphical tail-to-head method works for up to four vectors in any directions. The calculation (component method) gives the same answer without a scale drawing.

If the vectors drawn tail to head end exactly where the first one started, there is no gap for a resultant to fill: the resultant is zero. For forces, a closed vector diagram means the net force on the object is zero. Three forces that keep an object in equilibrium always form a closed triangle, and any one of them is equal in size and opposite in direction to the resultant of the other two.

F₁ = 12 N east, F₂ = 9 N north and F₃ = 15 N at 36,87° south of west, drawn tail to head. The head of F₃ ends at the tail of F₁: a closed vector diagram, so the resultant is zero.

Resolving a vector into components

Resolving works in the opposite direction to finding a resultant. Any vector can be replaced by two perpendicular vectors, a horizontal x-component and a vertical y-component, that together have exactly the same effect as the original vector. The vector is the resultant of its components.

Rₓ = R cos θ

Rᵧ = R sin θ

A force of 80 N at 30° to the positive x-axis. Its components are Fₓ = 80 N × cos 30° = 69,28 N and Fᵧ = 80 N × sin 30° = 40 N.

Force and displacement. The same method resolves any vector. A suitcase pulled with a force along a handle that slopes upwards moves because of the horizontal component of the pull, while the vertical component lifts some of its weight. A displacement of 15 km at 40° north of west has a westward component of 15 cos 40° = 11,49 km and a northward component of 15 sin 40° = 9,64 km.

A displacement of 15 km at 40° north of west. The angle is measured from the west side of the x-axis, so the x-component is −11,49 km (west) and the y-component is +9,64 km (north).

A force is a push or a pull, measured in newtons (N). In mechanics you will meet five kinds of force again and again: weight, the normal force, friction, applied forces and tension.

The normal force acts perpendicular to the surface, whether the plane is horizontal or inclined. On a horizontal floor it points straight up; on an inclined plane it points away from the slope at right angles to it. The normal force is not always equal to the weight: on an incline it is smaller than the weight, and when a rope pulls a box upwards at an angle, part of the pull lifts the box and the normal force becomes smaller too.

Friction acts parallel to the surface and opposes the motion (or the tendency to move) of the object relative to the surface. It is caused by the tiny irregularities of the two surfaces interlocking, which impedes motion. People have long used friction: the first people to make fire did so by rubbing sticks together.

A force diagram of a crate pushed to the right on a rough floor: the applied force F, friction f along the floor opposing the motion, the weight w downwards and the normal force N perpendicular to the floor.

Static and kinetic friction

Static friction adjusts itself. When you push a heavy crate gently and it does not move, the static frictional force is exactly as large as your push. Push harder and static friction grows to match, but only up to a maximum value, the maximum static frictional force f_s^max, reached just before the crate starts to move:

f_s^max = μ_sN

If the applied force exceeds f_s^max, the object starts to move and a net force accelerates it. Once it slides, kinetic friction acts. For a given pair of surfaces the kinetic frictional force is constant:

f_k = μ_kN

μ_s and μ_k are the coefficients of static and kinetic friction. They have no unit and depend only on the two surfaces. For most surfaces μ_k is smaller than μ_s, which is why it is easier to keep a crate sliding than to get it started. Use these formulas for objects at rest or sliding on horizontal and on inclined planes: in each case, N is the normal force on that surface.

Applied forces and tension

An applied force is a push or a pull by a person or a machine, labelled F (or F_A). Tension, T, is the pulling force in a string, rope or cable. A string always pulls: it pulls on the objects at both of its ends, along the string. In these problems the strings are light (their mass is negligible) and inextensible (they do not stretch), so the tension is the same all along the string and objects joined by it have the same acceleration.

Before any calculation with forces, draw the forces. Two kinds of drawing are used.

The free-body diagram of the pushed crate from the previous section. Only forces that act ON the crate are drawn; the arrow lengths show the relative magnitudes (here N = w and F is larger than f, so the crate accelerates).

Resolving forces on an inclined plane

On an inclined plane, choose the x-axis parallel to the slope and the y-axis perpendicular to it. The normal force and friction then lie along the axes, and only the weight has to be resolved. The angle between the weight and the perpendicular to the slope equals the angle of the incline, θ, so:

w_∥ = mg sin θ (parallel to the slope, pointing down it)

w_⊥ = mg cos θ (perpendicular to the slope, into it)

A block on an incline: its weight w and the components w∥ down the slope and w⊥ into the slope. The normal force N balances w⊥, so on a plain incline N = mg cos θ.

The net force in the x-direction is the vector sum of all the force components in the x-direction, and the net force in the y-direction is the vector sum of all the components in the y-direction. On an incline, nothing accelerates the block into or out of the slope, so the perpendicular components always add up to zero.

A force applied at an angle is resolved in the same way: a pull F at θ above the horizontal has a horizontal component F cos θ and a vertical component F sin θ that lifts part of the weight, so the normal force becomes N = mg − F sin θ.

A block pulled by a force F at an angle above the horizontal. The vertical component of F lifts part of the weight, so N is smaller than w.

Newton's first law and inertia

An object at rest stays at rest, and an object that moves keeps moving in a straight line at a constant speed, unless a net force acts on it. Both situations are called equilibrium: the net force is zero. Objects do not need a force to keep moving; they need a net force to change their motion (to speed up, slow down or change direction).

Mass is the measure of an object's inertia. The larger the mass, the larger the inertia and the harder it is to change the object's motion: a loaded truck is much harder to start, stop or turn than a bicycle. Newton's first law is therefore also called the law of inertia.

Why wearing seatbelts matters

In a collision a car stops in a very short time. According to Newton's first law, a passenger who is not wearing a seatbelt continues to move forward at the car's original velocity, because no force acts on them to stop them with the car. They keep moving until something stops them: the steering wheel, the dashboard, the windscreen or the road. A seatbelt supplies the unbalanced (net) force that slows the passenger down together with the car, spread over a longer time and over the strong parts of the body. Headrests work the same way in a collision from behind, where the body is pushed forward and the head tends to stay behind.

Newton's second law

F_net = ma

F_net is the net (resultant) force in newtons, m the mass in kilograms and a the acceleration in m·s⁻². One newton is the net force that gives a mass of 1 kg an acceleration of 1 m·s⁻²: 1 N = 1 kg·m·s⁻². The acceleration is always in the direction of the net force.

Objects in equilibrium and accelerating objects

Draw force diagrams and free-body diagrams for objects that are in equilibrium (at rest or moving with constant velocity) and for objects that are accelerating (non-equilibrium). For an object in equilibrium, all the forces along the plane of the motion add up to zero, and so do all the forces perpendicular to it. For an accelerating object, the arrows in the direction of motion do not balance: the longer arrow shows the direction of the net force.

Every problem in this section is solved the same way: a free-body diagram for each object, a positive direction, and F_net = ma applied to each object in each direction.

A single object on a horizontal or inclined plane

On a horizontal plane the normal force balances the weight (unless another force has a vertical component), and F_net along the plane is the applied force minus friction. On an inclined plane, take the direction of motion along the slope as positive. For a block sliding down a frictionless incline, the only force along the slope is w_∥:

mg sin θ = ma so a = g sin θ

On a rough incline, kinetic friction acts up the slope, against the motion, and N = mg cos θ, so f_k = μ_kmg cos θ. If the block is pushed or pulled up the slope, both w_∥ and friction act down the slope. A block at rest on an incline has static friction up the slope equal to w_∥.

A block sliding down a rough incline: friction f acts up the slope, opposite to the motion.

Vertical motion: lifts, rockets and apparent weight

For vertical motion, take the direction of the acceleration as positive. A lift cable pulls up with tension T; a rocket engine pushes up with a thrust. For a person standing on a scale in a lift, the scale reads the normal force it exerts on the person. This reading is the apparent weight, which differs from the true weight mg whenever the lift accelerates:

• Accelerating upwards (or slowing down while moving down): N − mg = ma, so N = m(g + a) and the person feels heavier.
• At rest or moving at constant velocity: N = mg; the apparent weight equals the weight.
• Accelerating downwards (or slowing down while moving up): mg − N = ma, so N = m(g − a) and the person feels lighter.
• In free fall (a = g) the normal force is zero and the person feels weightless.

Two-body systems

When two objects are joined by a light inextensible string, they move together with the same magnitude of acceleration, and the tension is the same at both ends of the string. Draw a free-body diagram for EACH body and apply Newton's second law to EACH body separately. This gives two equations, which are solved together for the acceleration and the tension. The four arrangements in CAPS are:

• Both on a flat horizontal plane, with or without friction.
• One on a horizontal plane and one hanging vertically from a string over a frictionless pulley.
• Both on an inclined plane, with or without friction.
• Both hanging vertically from a string over a frictionless pulley.

Both on a horizontal plane: F pulls the front block, and the string pulls the rear block along.

One on a horizontal table, one hanging over a frictionless pulley: the weight of the hanging block drives the system.

Both on an inclined plane: F pulls the upper block up the slope, and the string pulls the lower block.

Both hanging over a frictionless pulley: the heavier block accelerates down and the lighter block up.

Combining Newton's second law with the equations of motion

Many exam questions first ask for the acceleration from F_net = ma and then use it in an equation of motion, or work the other way: find the acceleration from the motion, then the force. The acceleration is the link between the two. The equations of motion hold only while the acceleration is constant, so the net force must be constant too.

v_f = v_i + aΔt

Δx = v_iΔt + ½aΔt²

v_f² = v_i² + 2aΔx

Δx = (v_i + v_f2)Δt

Forces always come in pairs, called action-reaction pairs. If you push on a wall, the wall pushes back on you just as hard. Neither force comes first: they act at the same time.

Identifying action-reaction pairs

Describe each force in the form "A exerts a force on B". Its pair is then "B exerts a force on A".

Other examples: a swimmer pushes the water backwards and the water pushes the swimmer forwards; a rocket pushes exhaust gases down and the gases push the rocket up.

F = Gm₁m₂r²

F is the gravitational force (in N) that each mass exerts on the other, m₁ and m₂ are the masses (in kg), r is the distance between their centres (in m) and G is the universal gravitational constant, G = 6,67 × 10⁻¹¹ N·m²·kg⁻². The forces on the two masses are equal in size and opposite in direction: they are an action-reaction pair.

r is measured from centre to centre, not from surface to surface.

Because G is so small, the force between everyday objects is tiny and cannot be felt. It becomes large only when at least one of the masses is enormous, such as a planet.

Weight and gravitational acceleration

w = mg

At the Earth's surface, the gravitational force on an object of mass m is its weight. Setting mg = GMmR² gives the gravitational acceleration at the surface of a planet of mass M and radius R:

g = GMR²

With the Earth's mass M = 5,98 × 10²⁴ kg and radius R = 6,38 × 10⁶ m, g = 9,8 m·s⁻²: the value used near the Earth. The same formula gives g on any planet from its own mass and radius, and w = mg then gives an object's weight on that planet.

Astronauts in an orbiting spacecraft are not beyond the reach of gravity: at that height g is still almost 90% of its value at the surface. They feel weightless because they and the spacecraft are falling around the Earth together, so the floor exerts no normal force on them. You feel the same sensation for a moment in a lift whose cable snaps, or at the top of a fast roller-coaster hill.