11.1.9.1: Illustrations

Illustration 1: Newton's First Law and Reference Frames

On first glance it may seem like Newton's first law (an object at rest remains at rest and an object in motion remains in motion unless acted on by a net force) is contained within Newton's second law. This is actually not the case. The first law is also a statement regarding reference frames. This is the information NOT contained in the second law. Sometimes the first law is also called the law of inertia. It defines a certain set of reference frames in which the first law holds, and these reference frames are therefore called inertial frames of reference. Put another way, Newton's first law states that if the net force on an object is zero, it is possible to find at least one reference frame in which that object has no acceleration.

A ball popper on a cart (not shown to scale) is shown moving on a track in five different animations (position is given in meters and time is given in seconds). In each animation, the ball is ejected straight up by the popper mechanism at \(t = 1\text{ s}\). Restart.

Let us first consider Animation 1. In this animation the cart is stationary. But is it really? We know that we cannot tell if we are stationary or moving at a constant velocity (in other words in an inertial reference frame). Recall that if we are moving relative to Earth at a constant velocity, we are in an inertial reference frame. So how can we tell if we are moving? How about the cart? We cannot tell if there is motion as long as the relative motion with respect to Earth can be described by a constant velocity. In Animation 1 the cart could be stationary. In this case, we expect—and actually see—that the ball lands back in the popper. However, if the cart was moving relative to Earth, and we were moving along with the cart, the motion of the ball and the cart would look exactly the same!

What would the motion of this ball and cart look like if it moved relative to our reference frame (or if we move relative to its reference frame)? Animations 2, 3, 4, and 5 show the motion from different reference frames.

First look at Animations 2 and 3. What do these animations look like? Both animations resemble projectile motion. The motion of the ball looks like motion in a plane as opposed to motion on a line. Does the ball still land in the popper? Would you expect this? Sure. There is nothing out of the ordinary going on here. Since there are no forces in the \(x\) direction, the motion of the ball (and cart) should be described by constant velocity in that direction. Therefore the ball and the cart have the same constant horizontal velocity.

Now look at Animations 4 and 5. What do these animations look like? Neither animation resemble projectile motion. The motion of the ball and cart look like they are being accelerated to either the right or the left (depending on the animation). Notice that the ball still lands in the popper as seen from this reference frame. Why do the ball and cart accelerate? There is nothing that we can see to explain why the ball and cart accelerate. Since Newton's laws must be correct, we must invent a force to describe why the cart and ball are accelerating (a fictitious force, since it really does not exist). Animations 4 and 5 depict motion as seen from noninertial reference frames.

Illustration 2: Reference Frames

In the two animations, we are given an example of a moving reference frame relative to Earth's (stationary) reference frame. The motion of the orange ball as seen in Earth's reference frame is depicted in the animation by the time, position, and velocity measurements, \(t,\: x_{1}\), and \(v_{1}\), respectively (position is given in meters and time is given in seconds). An observer is in another reference frame that is moving with a constant velocity with respect to the surface of Earth. The observer also takes down time, position, and velocity measurements as shown in the table and represented by \(t,\: x_{2}\), and \(v_{2}\) respectively. Animation 1 shows position and Animation 2 shows velocity. Restart.

How do we know that the observer in frame two is moving with respect to Earth's reference frame? At \(t = -2\text{ s}\), the observer in Earth's frame sees the orange ball at \(-4\text{ m}\) and moving to the right at a constant velocity of \(2\text{ m/s}\). What does the observer on the other reference frame see? She sees the ball start at the same position, but the ball moves with a different velocity in her frame. She sees it move to the right with a velocity of \(3\text{ m/s}\). Therefore, relative to Earth, our observer in frame 2 is moving with a velocity of \(1\text{ m/s}\).

But in what direction does the observer move? Consider the following question first. What if the observer-in her frame of reference-saw the ball as stationary? We would conclude that the observer was traveling at the same velocity as the ball as seen from the reference frame of Earth. When we move in the direction of the motion of the ball, the ball's relative velocity decreases. Thus, when we move in a direction opposite to the motion of the ball, the ball's relative velocity increases. Therefore the observer is moving to the left, relative to the reference frame of Earth, at \(1\text{ m/s}\)!

When a reference frame is moving uniformly (at a constant velocity) with respect to a non-accelerating (inertial) reference frame, the moving frame of reference is also called an inertial reference frame.

Illustration 3: The Zero-Momentum Frame

Is physics different when viewed in different reference frames? Well, it can certainly look different. Consider the collision in the animation as seen initially in the reference frame of Earth (the relative velocity between this frame and Earth's stationary frame is zero). Here both the red ball and the blue ball have the same mass equal to \(1\text{ kg}\). Note that energy and momentum are conserved in the collision with kinetic energy \(= 2\text{ J}\) and \(p_{x} = 2\text{ kg}\cdot\text{m/s}\) before and after the collision. Restart.

Change the velocity from zero to \(2\text{ m/s}\) (position is given in meters and time is given in seconds). How does the collision change? The red ball is now initially stationary and the blue ball is moving to the left at \(2\text{ m/s}\). Note that, in the original collision with \(v = 0\text{ m/s}\), the red ball was initially moving to the right and the blue ball was initially stationary. In the new frame the momentum of the two-ball system is different. However, the kinetic energy happens to be the same and energy and momentum are conserved.

Now try \(v = -2\text{ m/s}\). Are energy and momentum still conserved? Even though the values of the kinetic energy and momentum change, the laws of conservation of energy and conservation of momentum still hold.

Now try \(v = 1\text{ m/s}\). What is the new momentum for the two-ball system? This frame of reference is appropriately called the zero-momentum frame. In this frame the momentum of the system is zero. This frame is also called the center-of-mass frame. The center of mass is a coordinate that is a mass-weighted average of the positions of the objects that make up the system. In a two-object system the center of mass is always somewhere in between the two objects. Since the center of mass is a mass-weighted average, the center of mass will always be closer to the object that is more massive. In the case of this animation, where both balls have the same mass, the center of mass is always at the midpoint between the two masses. This point does not move in the zero-momentum frame, but does move in other frames.

Illustration 4: Rotating Reference Frames

In an inertial reference frame, momentum and energy are conserved even though observers may disagree on total momentum or total energy. As a consequence, if two observers were in inertial reference frames, the two observers would agree on the forces acting. Restart.

Consider a green mass at the end of a spring (position is shown in meters and time is shown in seconds). If the mass is not moving, the net force on it is zero, and therefore the gray shell represents the equilibrium position of the spring.

Now imagine that someone has given the green ball a brief push making it rotate with a constant speed as seen in the laboratory frame. We know that the spring must be stretched as shown. Why? Well, we need a force toward the center of the circle because this force is necessary for uniform circular motion. This force is the spring force.

Imagine you (the woman in the animation) are riding on the green mass. From your point of view, mass's reference frame, what is the motion of the green ball? It is stationary. In this reference frame the ball does not move. It is not, however, an inertial reference frame because it is accelerating. What happens to the spring in this frame? The spring is stretched from its equilibrium position as before. How would you explain this? From your point of view riding with the rotating mass, since you are not accelerating, the net force on the green ball is zero, and someone or something has stretched the spring by pulling it outward. This force is purely a figment of your imagination. This fictitious force—also called the centrifugal force (no more real because it has a name!)—must be invented by an observer on the green ball if she is going to keep Newton's laws as a fact of Nature. Whenever you are in a rotating reference frame, like a merry-go-round, for example, you are in an accelerating reference frame, and therefore you must invent fictitious forces to make Nature obey Newton's laws.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.