Assume you are sitting in your airplane seat, and the airplane moves with constant velocity v through smooth air.  Your tray is pulled down and a glass filled with juice is sitting on it.  The surface of the liquid is perfectly smooth.  Are the glass and the liquid at rest?

• The glass and the liquid are at rest in the reference frame of the airplane.   If the origin of a coordinate system is fixed in the cockpit of the airplane, and the x-, y-, and z-axes intersect there, then the x-, y-, and z-coordinates of the center of the glass do not change with time.
• The glass and the liquid, however, are not at rest in the reference frame of an observer on the ground.  If the origin of a coordinate system is fixed on the ground and the x-axis points into the direction of travel of the plane, then the x-coordinate of the center of the glass changes as x(t) = x₀ + vt.  In the reference frame fixed on the ground the airplane moves with constant velocity v.

The state of motion (i.e. the velocity) of any object is always defined with respect to a reference frame.

You are getting thirsty and decide to have a drink.  You grab the glass, pull it towards you, and lift it towards your lips.  The state of motion of the glass changes in both reference frames.  If something pushes or pulls on an object, we say that a force is acting on the object.  A force is a vector.  It has magnitude and direction.  If two or more forces act on an object, then the net force acting on the object is the vector sum of all the forces.

If we only consider forces in one dimension, say, along the x-axis, then the sign of the force F functions as the direction indicator.  A negative F points into the negative x-direction and a positive F points in the positive x-direction.  To find the net force, we just add the numbers.

In one dimension only:  F_net = F₁ + F₂ + F₃ + ...  = ∑F_i.
(The symbol ∑ stands for the sum.)

You pull on the glass, therefore you are exerting a force on the glass.  The glass is now no longer at rest in the frame of the airplane, and it is no longer moving with constant velocity in the reference frame fixed on the ground.  You apply a force to the glass, and the glass changes its velocity as seen in both frames.  The glass is accelerating in both frames.

Newton's first law, also called the law of inertia, defines a special class of reference frames, called inertial frames.  It states that, when viewed in an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with constant velocity unless it is acted on by an external net force.  The frame fixed in the airplane moving with constant velocity v and the frame fixed on the ground are inertial frames.

Question:

Are all frames inertial frames?

• No!  Assume the pilot sees another plane on a collision course, and he initiates evasive maneuvers.  The airplane suddenly accelerates towards the right.  Your seatbelt holds you firmly in your seat, but the glass and the juice slide to the left and fall off the tray.  No net force is acting on the glass, but in the reference frame fixed in the airplane the glass is accelerating.  The maneuvering airplane is not an inertial frame.

Accelerating reference frames are never inertial frames.

Question:

Is a reference frame with its origin fixed at a point on the surface of the earth an inertial frame?

• No!  This frame is an accelerating frame, and therefore not an inertial frame.  When viewed from space, the point on the surface of the earth is accelerating.  It moves in a circle as the earth spins on its axis.  With the earth, it also orbits the sun, and with the solar system it orbits the center of the galaxy.  However, the magnitude of the acceleration of this frame is usually very small when compared to the magnitude of other relevant accelerations, and we often treat this frame as an inertial frame when solving problems.
  The radius of the earth is r = 6368 km.  The earth rotates about its axis once every 24 hours.  A point on the surface of the earth on the equator moves with speed v = 2πr/(24 h) = 463 m/s.  The magnitude of its acceleration is a = v²/r = 0.034 m/s².  This is very small compared to the gravitational acceleration, which is 9.8 m/s².

In an inertial frame the state of motion of an object only changes if there is a net force acting on the object.  If there is no net force acting on an object, its velocity will not change.  If it is initially at rest, it will stay at rest, if it is moving with a given speed in a certain direction, it will keep on moving with the same speed in the same direction.

The merry-go-round in the video clip below is not an inertial frame.  In this frame the ball does not move in a straight line with constant velocity, even though the net force on the ball is zero.  Play the video clip or step through it frame by frame.

Or watch the clip on Youtube.

Links:

• The car and the wall
• The motorcyclist

Problem:

In the motion picture "It Happened One Night" (Columbia Pictures, 1934), Clark Gable is standing inside a stationary bus, in front of Claudette Colbert, who is seated.  The bus suddenly starts moving forward, and Clark falls into Claudette's lap.  Why did this happen?

Solution:

• Reasoning:
  The actor and the bus are initially at rest in an inertial frame fixed on the ground.  A force causes the bus to accelerate forward.  No force in the horizontal direction is initially acting on the actor, so he will remain at rest with respect to the ground.  Once the bus starts moving, frictional forces between the floor of the bus and the actor's feet will try to prevent relative movement.  Since relative to the floor of the bus the actor moves backward, the frictional forces opposing this motion pull his feet in the forward direction.  His feet do not stay under his center, and he falls.

Module 4: Question 1

What is wrong with the statement "Because the car is at rest, there are no forces acting on it."?  How would you correct this sentence? 

Discuss this with your fellow students in the discussion forum!

Link:  Newton's first law of motion