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Forces and Newton’s Laws

Physics · Forces and Newton’s Laws

Newton’s Second Law

The video lesson for this topic is coming soon.

Essential Question

Why is a full shopping cart harder to speed up than an empty one?

Give an empty cart a shove and it zooms off. Give a full cart the same shove and it barely moves. By the end of this lesson, you should be able to explain how force and mass affect acceleration and use Fnet = ma to solve for any of the three.

Force Causes Acceleration

  • Net force causes acceleration
  • a points the same way as Fnet
  • No net force, no acceleration
A cart on the ground. A blue arrow starts at the cart's center and points right, labeled F_net. Above the cart, a gray arrow also points right, labeled a. Caption: acceleration points with the net force.

Newton’s first law says an unbalanced force changes an object’s motion. The second law tells us how. A net force makes an object accelerate, and the acceleration always points in the same direction as the net force. If the net force is zero, the acceleration is zero and the velocity stays the same.

Force and Acceleration

  • More force, more acceleration
  • Double Fnet, double a
  • Mass stays the same
A graph of acceleration in meters per second squared versus net force in newtons. The data points fall on a straight line through the origin. Caption: straight line through the origin means directly proportional.

If you keep the mass the same and increase the net force, the acceleration increases by the same factor. Double the force and you double the acceleration. On a graph of a versus Fnet, this makes a straight line through the origin. We say acceleration is directly proportional to net force.

Mass and Acceleration

  • More mass, less acceleration
  • Double m, half the a
  • Force stays the same
A graph of acceleration in meters per second squared versus mass in kilograms. The curve starts high and drops, flattening out as mass increases. Points at 2 kg, 4 kg, and 8 kg show the acceleration halving each time the mass doubles. Caption: curve that drops: inverse.

If you keep the net force the same and increase the mass, the acceleration goes down. Double the mass and the acceleration is cut in half. With a 12 N net force, a 2 kg cart accelerates at 6 m/s2, a 4 kg cart at 3 m/s2, and an 8 kg cart at 1.5 m/s2. We say acceleration is inversely proportional to mass.

Newton’s Second Law

a = Fnet / m

  • a = acceleration, in meters per second squared (m/s2)
  • Fnet = net force, in newtons (N)
  • m = mass, in kilograms (kg)
The equation a = F_net / m in a yellow box. Below it, three cards: a is acceleration in meters per second squared, F_net is net force in newtons, and m is mass in kilograms.

Newton’s second law puts both relationships into one equation. Net force is on top, so more force means more acceleration. Mass is on the bottom, so more mass means less acceleration. Always use the net force, not just one of the forces.

Solving for Any Variable

  • Find a: a = Fnet/m
  • Find force: Fnet = ma
  • Find mass: m = Fnet/a
A triangle split into three parts with F_net on top and m and a on the bottom. Caption: cover the one you want.

Newton’s second law is usually written as Fnet = ma. Rearrange it to solve for whichever quantity the problem asks for. With the triangle, cover the quantity you want: covering Fnet leaves m next to a, so multiply. Covering m leaves Fnet over a, so divide.

The Newton

  • 1 N = 1 kg·m/s2
  • Gives 1 kg an a of 1 m/s2
  • Units check your work
A block labeled 1 kg on the ground is pulled to the right by a black arrow labeled 1 N. A gray arrow above it is labeled a = 1 m/s². Caption: 1 N on 1 kg gives 1 m/s².

Newton’s second law defines the newton. One newton is the net force that gives a 1 kg mass an acceleration of 1 m/s2. Since F = ma, the units multiply too: 1 N = 1 kg·m/s2. Checking that your units work out is a great way to catch mistakes.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Finding Acceleration

A net force of 36 N east acts on a 12 kg wagon. What is the wagon’s acceleration?

A wagon labeled 12 kg with a blue arrow pointing east labeled F_net = 36 N.
  1. Knowns: Fnet = 36 N east, m = 12 kg
  2. Equation: a = Fnet/m
  3. Substitute: a = 36 N / 12 kg = 36 kg·m/s2 / 12 kg
  4. Answer: a = 3.0 m/s2 east

Example 2: Finding Net Force

A 75.0 kg sprinter accelerates out of the starting blocks at 4.20 m/s2 forward. What net force acts on her?

A runner leaning forward with a gray arrow pointing forward labeled a = 4.20 m/s² and a label m = 75.0 kg.
  1. Knowns: m = 75.0 kg, a = 4.20 m/s2 forward
  2. Equation: Fnet = ma
  3. Substitute: Fnet = (75.0 kg)(4.20 m/s2) = 315 kg·m/s2
  4. Answer: Fnet = 315 N forward

Example 3: Finding Mass

A net force of 24 N gives a bowling ball an acceleration of 4.0 m/s2. What is the ball’s mass?

A bowling ball on the ground with a blue arrow labeled F_net = 24 N and a gray arrow labeled a = 4.0 m/s².
  1. Knowns: Fnet = 24 N, a = 4.0 m/s2
  2. Equation: Fnet = ma, so m = Fnet/a
  3. Substitute: m = 24 N / 4.0 m/s2 = 24 kg·m/s2 / 4.0 m/s2
  4. Answer: m = 6.0 kg

Example 4: Using a Free-Body Diagram

A 15 kg box is pushed east with a 92 N force. Friction acts on it with 47 N west. What is the box’s acceleration?

A box labeled 15 kg on the floor. A black arrow labeled 92 N pushes it east, and a green arrow labeled 47 N points west.
  1. Net force: east is +: Fnet = Fapp − Ff = 92 N − 47 N = 45 N east
  2. Equation: a = Fnet/m
  3. Substitute: a = 45 N / 15 kg
  4. Answer: a = 3.0 m/s2 east

Using only the 92 N push would give 6.1 m/s2, which is wrong. Newton’s second law uses the net force.

Big Ideas

  • A net force causes acceleration in the same direction
  • a = Fnet/m: more force, more a; more mass, less a
  • Rearrange: Fnet = ma, m = Fnet/a
  • 1 N = 1 kg·m/s2

Back to the essential question: a full cart has more mass, and acceleration equals net force divided by mass, so the same push gives a full cart a smaller acceleration. More force means more acceleration, and more mass means less. Next up, we’ll look at the difference between mass and weight.

Take It With You

Printable lesson slides and a practice worksheet for this topic are coming soon.