Newton's laws of motion

What statics is about

Engineering mechanics uses Newton's physics to predict how structures and machines respond to the forces on them. Statics is its first branch: the study of bodies in equilibrium, meaning bodies that are not accelerating. In practice they are usually at rest, which is where the name comes from. The ideas are old. Archimedes stated the law of the lever in the third century BCE, Simon Stevin analysed loads on inclined planes in 1586, and Newton's laws of 1687 gave the whole subject a single foundation. Statics leads on to dynamics, mechanics of materials and fluid mechanics, and you will use it in every one of them.

Translation and rotation

A body can move in two basic ways. In translation it changes position without changing its orientation: a drawer sliding straight out of a desk translates. In rotation it turns about an axis: a door swinging on its hinges rotates. Many motions combine both, such as a car wheel rolling along a road, which moves forwards while it spins. The useful fact is that translation and rotation can be analysed separately, with separate equations. Forces govern translation, and moments, the turning effects of forces, govern rotation. Statics needs both.

Newton's first law

The first law says that a body stays at rest, or keeps moving in a straight line at constant speed, unless an unbalanced external force acts on it. It is often called the law of inertia. Because velocity is a vector, an unbalanced force is needed to speed a body up, slow it down or change its direction. The same idea applies to turning. A flywheel left to spin would keep spinning at the same rate if nothing resisted it; it eventually stops because friction at its bearings applies a small opposing moment. Engineers say moment where physicists usually say torque.

Newton's second law

The second law connects the net force on a body to its acceleration:

Bold symbols are vectors, so the acceleration points in the direction of the net force, and its size is proportional to the net force and inversely proportional to the mass \(m\), a scalar. The physics topic "Newton's second law and systems" develops this in full. For turning about a fixed axis there is a matching law, \(\Sigma M = I\alpha\): a net moment causes an angular acceleration \(\alpha\), resisted by the mass moment of inertia \(I\), which the topic "Products of inertia and mass moment of inertia" defines.

From the second law to equilibrium

In statics the body does not accelerate, so \(\mathbf{a} = \mathbf{0}\) and \(\alpha = 0\). The two laws then collapse to the conditions of equilibrium:

The forces on the body must balance, and so must their moments. Zero acceleration does not mean zero velocity: a load rising at a steady speed is in equilibrium just as surely as one at rest. Almost every later topic in this course is an application of these two vector equations, written out as scalar equations in each direction.

Worked example: a crate on a crane hook

A 450 kg crate hangs from a crane cable. Find the cable tension (a) while the crate is held still and (b) while it is lowered at a steady 0.3 m/s. Use the engineering value \(g = 9.81\) m/s², explained in the next topic. 1. In both cases the acceleration is zero, so \(\Sigma F = 0\) applies. 2. Isolate the crate: two forces act, its weight downward and the tension \(T\) upward. 3. The weight is

4. Taking up as positive, \(T - W = 0\), so \(T = 4.41\) kN in both cases. 5. Check: the steady motion changes nothing, because only acceleration requires an unbalanced force.

Newton's third law

The third law says that forces come in pairs. When body A pushes or pulls on body B, B pushes or pulls on A with a force of equal size in the opposite direction, at the same instant. This holds for contact forces, such as a foot on a floor, and for forces at a distance: Earth pulls a book down with the book's weight, and the book pulls Earth up with an equal force. The two forces of a pair always act on different bodies, so they never cancel each other in an analysis of one body. The physics topic "Newton's third law" covers more examples.

Free-body diagrams expose the pairs

Third-law pairs are easy to confuse when several bodies touch. The cure is the free-body diagram: a sketch of one body, or a group of bodies, cut free from everything around it, showing only the forces that act on it. Forces the body exerts on its neighbours are left off. If you draw two touching bodies separately, each interaction force appears once on each diagram, in opposite directions. If you draw them together as one system, the pair becomes internal and drops out. The topic "Equilibrium, particles, and free-body diagrams" turns this into a full method.

Worked example: a motor on a workbench

A 12 kg motor sits on a 30 kg workbench standing on a floor. Find the force between motor and bench, and the total floor force on the bench. 1. Motor alone: its weight, \(12 \times 9.81 = 117.7\) N, acts down, so the bench must push up on it with 117.7 N. 2. Third law: the motor pushes down on the bench with 117.7 N. 3. Bench alone: its weight is \(30 \times 9.81 = 294.3\) N, plus the motor's 117.7 N push, both down, so the floor pushes up with 412.0 N in total. 4. Check with motor and bench as one system: the 117.7 N pair is internal, and the floor supports \(42 \times 9.81 = 412.0\) N. The answers agree.

Common mistakes to avoid

First, thinking that a moving body needs a net force to keep moving: steady motion needs none. Second, cancelling a third-law pair on one free-body diagram, when the two forces act on different bodies. Third, calling weight and the normal force a third-law pair: both act on the same body and are different kinds of force; the partner of the weight is the body's pull on Earth. Fourth, checking only \(\Sigma \mathbf{F} = \mathbf{0}\) and forgetting that equilibrium also needs \(\Sigma \mathbf{M} = \mathbf{0}\).

Key takeaways

Statics studies bodies in equilibrium, which do not accelerate. Motion can be translation, rotation or both, and each is analysed separately. The first law says a body keeps its velocity unless an unbalanced force acts; moments play the same role for rotation. The second law, \(\Sigma \mathbf{F} = m\mathbf{a}\), with \(\Sigma M = I\alpha\) for rotation, reduces in statics to \(\Sigma \mathbf{F} = \mathbf{0}\) and \(\Sigma \mathbf{M} = \mathbf{0}\). The third law pairs every force with an equal and opposite force on another body, and free-body diagrams keep the pairs straight.

Question 1 of 5

A lift car rises at a steady 2 m/s. What is the net force on it?

Question 2 of 5

Which conditions must hold for every body in static equilibrium?

Question 3 of 5

A 450 kg crate is lowered on a crane cable at a constant 0.3 m/s. Using g = 9.81 m/s², what is the cable tension?

Question 4 of 5

A book rests on a table. What is the third-law partner of the book's weight?

Question 5 of 5

Which of these motions is pure translation?