Newton’s Second Law of Motion

Force saves you from the monotony of everything moving at the same speed and direction forever. Force can act on objects, changing their direction and/or speed. The relationship between force, mass, and acceleration is primary in physics classes.

To start, you need to know Newton’s Second Law of Motion, which is a big one in physics: “When a net force sigmaF acts on an object of mass m, the acceleration of that mass can be calculated by sigmaF = ma.” The sigma sign stands for “sum,” so sigmaF = ma can be read as “the vector sum of all forces on an object (the net force) equals mass times acceleration.” It is important to note that this is a vector equation and that both force and acceleration are vectors in this equation. Mass, like time, is a scalar.

Newton’s second law is often just abbreviated as F = ma. It’s important to note that sigmaF refers to the vector sum of all the forces on the object and not to just a single force.

What is the unit of force? This table gives you a rundown for the three measurement systems most commonly used in physics:

Here’s how you relate the three different units of force:

1 lb = 4.448 N

1 N = 10⁵ dyn

Sample question

1. You’re at rest on an ice rink when you get hit from behind with a force of 50.0N as someone bumps you. If your mass is 70.0 kg, what is your acceleration?

   The correct answer is 0.714 m/s².

   1. Use the equation F = ma: Solving for a gives you a = F/m.

   2. Plug in the numbers: a = F/m = (50.0 N)/(70.0 kg) = 0.714 m/s².

Practice questions

1. You come home to find a delivered package with a mass of 100 kg blocking the door. If you push it with a force of 100 N, what will its acceleration be if no friction is involved?

2. You’re gliding across a frictionless lake in a sailboat. If your mass is 70 kg and the boat’s mass is 200 kg, with what force does the wind need to blow you to give you an acceleration of 0.30 m/s²?

3. You have control of a space station, which has a mass of 400,000 kg. To give it an acceleration of 2.0 m/s², what force do you need to apply with the rockets?

4. You find a stone in the forest and give it a push of 50.0 N. It accelerates at 2.0 m/s². What is its mass?

5. You’re applying a force of 17 N to a hockey puck with a mass of 0.17 kg. Starting from rest, how far has the puck gone in 0.10 seconds?

6. You push a rowboat on a calm lake (assuming no friction) with a force of 40.0 N. If the rowboat has a mass of 80.0 kg, how far has it gone in 10.0seconds?

7. A space station with a mass of 1.0 x 10⁵ kg is moving toward a satellite at 5.0 m/s. If you want to avoid crashing them together and have only 1.0 x 10³ seconds in which to act, what force do you need to apply to stop the space station from colliding with the satellite?

8. Your 1,000-kg car needs a push. Starting at rest, how hard do you have to push to get it up to 5.0 m/s in 1.0 x 10² seconds?

Following are answers to the practice questions:

1. 1 m/s²

   1. Solving F = ma for a gives you a = F/m.

   2. Plug in the numbers: a = F/m = 100 N/100 kg = 1 m/s².

2. 81 N

   1. Use the equation F = ma.

   2. Plug in the numbers: F = ma = (70 kg + 200 kg)(0.30 m/s²) = 81 N.

3. 800,000 N

   1. Use the equation F = ma.

   2. Plug in the numbers: F = ma = (400,000 kg)(2.0 m/s²) = 800,000 N.

4. 25 kg

   1. Use the equation F = ma, and solve for the mass, giving you m = F/a.

   2. Plug in the numbers: m = F/a = 50 N/2.0 m/s² = 25 kg.

5. 50 m

   1. Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

   2. Use the equation

      

      and substitute F/m for a:

      

   3. Plug in the numbers:

      

6. 25 m

   1. Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

   2. Use the equation

      

      and substitute F/m for a:

      

   3. Plug in the numbers:

      

7. –5.0 x 10² N

   1. Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

   2. Use the equation v_f = v_o + at. In this question, v_f = 0, so at = –v_o.

   3. This becomes

      

      solving for F gives you

      

   4. Plug in the numbers, and you get F = –500 N = –5.0 x10² N (it’s negative because it’s opposite to the direction of travel).

8. 50 N

   1. Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

   2. Use the equation v_f = v_o + at. In this question, v_f = 10 and v_o = 0, so at = v_f.

   3. This becomes

      

      solving for F gives you

      

   4. Plug in the numbers, and you get F = 50 N.