Newton’s first law defines a class of inertial frames. Inertial frames are reference frames for which the trajectories for force-free motion are solutions to dr/dt = 0. With respect to inertial frames Newton’s second law has the form
F = dp/dt. (r = coordinate, F = force, p = mv momentum)
Let Fik be the force that particle i exerts on particle k. Newton’s third law states that Fik = -Fki.
Newton’s laws are well suited for the study of unconstrained mechanical systems. Constraints, such as requiring a particle to follow a given curve in space, tell us that there are external forces, but do not tell us what these forces are. The forces are only known in terms of their effect on the motion.
Conservation laws are very important tools in solving mechanics problems.
| For a system of particles momentum is conserved if Fext = 0; Fext = 0 Û P = constant. |
| Angular momentum (L = r´p) is conserved if the torque text = 0; text = 0 Û L = constant. |
| Energy E = T + U is conserved if all forces are conservative; òF× dr = 0 Û T + U = constant. |
| Laws: | |
| Newton's 2nd law: | F = dp/dt |
| Newton's third law: | Fik = -Fki |
| Forces: | |
| Static and kinetic friction: | fs £ msN, fk = mkN |
| Gravity: |
 |
| Uniform circular motion: | F = mv2/r |
| Hooke's law: | F = -kr, Fx= -kx |
| Concepts: | |
| Work: | W = F×d |
| Kinetic energy: | K = (1/2)mv2 |
| Work-kinetic energy theorem: | Wnet = DK = (1/2)m(vf2-vi2) |
| Elastic potential energy: | U = (1/2)kx2 |
| Gravitational potential energy: |  |
| Conservative systems: | E = K + U, Fx = -dU/dx |
| Power: | P = F·v or P = dW/dt |
| Momentum: | p = mv |
| Impulse: | I = Dp = FavgDt |
| Angular momentum: | L = r´p |
| Torque | t = r´F |
| Angular momentum and torque: |
 ,
 ,
dL = tdt |
Forces and torques that act so powerfully but so briefly that they produce finite changes in linear and angular momentum while the system undergoes negligible displacement are said to be impulsive.
Linear Impulse: dp = Fdt, Dp
= òFdt, Dp
= FavgDt
The integral of force over time as Dt approaches 0
is called the impulse
of the force.
Angular impulse: dL =
tdt, DL
= òtdt, DL
= tavgDt
The integral of torque over time as Dt approaches 0
is called the angular impulse
of the torque.
In collisions, it is assumed that the colliding particles interact for such a short time, that the impulse due to external forces is negligible. Thus the total momentum of the system just before the collision is the same as the total momentum just after the collision.
| Elastic collision: momentum is conserved, mechanical energy is conserved |
| Inelastic collisions: momentum is conserved, mechanical energy is not conserved |