Lesson 1 · 20 min
Newton's Second Law
Kinematics describes motion without asking what causes it. Kinetics asks: which forces act on a body, and what motion do they produce? One law answers both directions of that question. The resultant force on a particle equals its mass times its acceleration, \(\sum\Fvec = m\avec\), and every problem in this module is that equation, written carefully for a well-drawn free-body diagram.
Learning objectives
- State Newton's three laws of motion and the equation of motion \(\sum\Fvec = m\avec\) for a particle.
- Explain what an inertial frame of reference is, and why the ground is one for engineering problems.
- Distinguish mass from weight, and use \(W = mg\) and the newton, \(1\ \text{N} = 1\ \text{kg·m/s}^2\).
- Recognize the three kinds of kinetics problem: forces to motion, motion to forces, and a mix of both.
A push on smooth ice
A crate rests on smooth ice. You push it horizontally with a steady force. Before you look at Figure 1.1, make a prediction.
Double the push and the velocity line gets twice as steep. Double the mass instead and it gets half as steep. Add friction and the slope drops by \(\mu_k g\); make the push smaller than the largest static friction and nothing moves at all: the forces balance, and the acceleration is zero.
Newton's laws of motion
- First law. A particle at rest stays at rest, and a particle in motion keeps moving in a straight line at constant speed, unless an unbalanced force acts on it.
- Second law. A particle acted on by an unbalanced force accelerates in the direction of the force, at a rate proportional to the force.
- Third law. The forces two particles exert on each other are equal in size, opposite in direction and collinear.
The equation of motion
\[ \textstyle\sum\Fvec = m\colMA{\avec} \]The resultant of all the forces on a particle equals its mass times its acceleration. Both sides are vectors, so the equation holds component by component.
- The first law is the special case \(\sum\Fvec = \mathbf 0\), \(\avec = \mathbf 0\): statics is dynamics with no acceleration.
- The third law is what lets us split a machine into parts: the force a cord exerts on a block is equal and opposite to the force the block exerts on the cord.
- "Particle" means a body whose size and rotation do not matter to the question. A car on a curve, a crate on a ramp or a satellite in orbit can all be treated as particles.
Mass, weight and units
Mass \(m\) is a property of the body: its resistance to acceleration. It is the same everywhere. Weight is the force of gravity on the body, and it depends on where the body is. Near the Earth's surface:
Weight
\[ \colW{W = mg}, \qquad g = 9.81\ \text{m/s}^2 \]In SI units, mass is in kilograms, acceleration in meters per second squared, and force in newtons. The newton is defined by the second law: \(1\ \text{N}\) gives \(1\ \text{kg}\) an acceleration of \(1\ \text{m/s}^2\). So a \(70\ \text{kg}\) person weighs \(70(9.81) = 686.7\ \text{N}\) on Earth, and about \(113\ \text{N}\) on the Moon, where \(g = 1.62\ \text{m/s}^2\), with the same mass.
A bathroom scale measures the force it pushes up with, not the mass. That is why it reads differently in an accelerating elevator (Lesson 2).
Example 1.1 — The force to accelerate a car
A \(1200\ \text{kg}\) car accelerates uniformly from rest to \(100\ \text{km/h}\) in \(8\ \text{s}\) on a level road. What net horizontal force acts on it?
Show solution
Kinematics first: \(v = 100/3.6 = 27.78\ \text{m/s}\), so \(a = 27.78/8 = 3.472\ \text{m/s}^2\). Then kinetics:
\[ \textstyle\sum F_x = ma = (1200)(3.472) = 4167\ \text{N} \]This is the net force: the push of the road on the driven wheels less the air drag and rolling resistance. It is about \(35\%\) of the car's weight, \(11.77\ \text{kN}\).
Three kinds of kinetics problem
| Given | Find | Example |
|---|---|---|
| The forces | The acceleration, then the motion by kinematics | How fast is a crate moving after it slides \(3\ \text{m}\) down a rough ramp? |
| The motion | The forces that cause it | What tension does an elevator cable carry while the car speeds up? |
| Some of each | The unknown forces and the acceleration together | Two blocks joined by a cord: find their acceleration and the tension. |
In every case the work is the same: a free-body diagram showing every force, a kinetic diagram showing \(m\avec\), and one equation for each direction. Lesson 2 sets out the procedure.
Color tells you which force is which
Every figure, readout and highlighted formula in this module uses the same colors.
- Weight\(\colW{W = mg}\), drawn ending at the center of gravity
- Normal forcesThe push \(N\) of a supporting surface, ending on the face it pushes
- Friction\(\colF{F_f}\), drawn thicker, starting on the face where it acts
- Applied forces and dragA push or pull \(\colP{P}\), or air resistance
- SpringsThe spring force \(\colS{F_s = ks}\)
- Mass times acceleration\(\colMA{m\avec}\), on the kinetic diagram, and the acceleration \(\colMA{\avec}\)
- Velocity\(\colV{\vvec}\)
Cord tensions are drawn in gray, starting where the cord is tied.
The road ahead
- Lesson 2 draws free-body and kinetic diagrams and sets out a procedure. Lesson 3 applies it in rectangular coordinates, with friction and inclines.
- Lesson 4 handles bodies joined by cords and pulleys, and Lesson 5 forces that change with time, speed or position.
- Lessons 6 and 7 write the law along and across a curved path: flat and banked curves, crests and loops.
- Lesson 8 applies the law to systems of particles and compares it with the energy and momentum methods of the modules that follow. Equations of motion in cylindrical coordinates are in the Cylindrical Coordinates module.
Check your understanding
Key takeaways
- \(\sum\Fvec = m\avec\): the resultant of every force on a particle equals its mass times its acceleration, measured in an inertial frame.
- No unbalanced force, no acceleration: the first law and statics are special cases.
- Mass is the same everywhere; weight is \(W = mg\) and depends on \(g\). \(1\ \text{N} = 1\ \text{kg·m/s}^2\).
- Kinetics problems go from forces to motion, from motion to forces, or both at once.
- Next, Lesson 2 turns the law into a procedure built on free-body and kinetic diagrams.