Newton’s Laws
First Law A body moves with constant velocity unless a net force acts on the body.
Second Law The rate of change of momentum of a body is equal to the net force applied to the body.
Third Law If object A exerts a force on object B, then object B exerts a force on object A. These have equal magnitude but opposite direction. Newton’s second law
The second law should be familiar:
F = ma where m is the inertial mass (a scalar) and a is the acceleration (a vector). Note that a is the rate of change of velocity, which is in turn the rate of change of displacement. So
d d2 a = v = r dt dt2 which, in simpli ed notation is
a = v˙ = r¨ The principle of relativity
The principle of relativity The laws of nature are identical in all inertial frames of reference
An inertial frame of reference is one in which a freely moving body proceeds with uniform velocity. The Galilean transformation
- In Newtonian mechanics, the concepts of space and time are completely separable. - Time is considered an absolute quantity which is independent of the frame of reference: t0 = t. - The laws of mechanics are invariant under a transformation of the coordinate system. u y y 0 S S 0
x x0
Consider two inertial reference frames S and S0. The frame S0 moves at a velocity u relative to the frame S along the x axis. The trans- formation of the coordinates of a point is, therefore
x0 = x − ut y 0 = y z0 = z
The above equations constitute a Galilean transformation u y y 0 S S 0
x x0
These equations are linear (as we would hope), so we can write the same equations for changes in each of the coordinates:
∆x0 = ∆x − u∆t ∆y 0 = ∆y ∆z0 = ∆z u y y 0 S S 0
x x0
For moving particles, we need to know how to transform velocity, v, and acceleration, a, between frames. As v = r˙, we have
0 vx = vx − u 0 vy = vy 0 vz = vz u y y 0 S S 0
x x0
The acceleration is the rate of change of velocity. The speed u of frame S0 is a constant, so
0 ax = ax 0 ay = ay 0 az = az So, the acceleration of a particle in one frame is the same in any inertial frame. Such a quantity is known as an invariant. Galilean invariance
There are three key quantities that remain the same (are invariant) under a Galilean transformation between inertial reference frames: - Time: t = t0 - Inertial mass: m = m0 - Acceleration: a = a0 We can see already from this that a Galilean transformation is going to preserve Newton's laws. . . Invariance of Newton’s second law
In frame S d d F = p = mv = ma dt dt We can see that the direction of acceleration is the same as the force and that |F | m = |a| is the inertial mass, i.e. the resistance of a body to it being accelerated. Note that only external forces can change the state of motion (I cannot lift myself, for example). In the frame S0 d F 0 = m (v − u) = mv˙ − mu˙ dt As u is a constant, u˙ = 0, and so
F 0(x0, y 0, z0) = F (x, y, z) which is what we expect from a vector: the force is the same but the individual components of the force change with a change of reference frame.
We will return to invariance later when we consider special relativity. For now, it is su cient that we know that forces do not depend on our frame of reference.