Kinematics

Newton’s laws of motion

Newton’s first law states that when the net force on a body is zero, the acceleration of the body will also be zero. This is implied by the second law.

Newton’s second law states that the net force on a body is equal to the body’s acceleration multiplied by its mass, which is equal to the rate at which the body’s momentum is changing with time:

\vec{F} = m\vec{a}

Newton’s third law states that if two bodies exert forces on one other, then these forces have the same magnitude but opposite directions.

Linear motion

Change in position is given either as distance (a scalar, distance covered by a course), or displacement (a vector, directed distance between start and end points). Change in position is given by \Delta d = vt.

Speed v is the distance travelled in a unit of time (the rate of change of position). Velocity \vec v is directed speed. Average speed or velocity are given by v = \frac{\Delta d}{t}. Change in velocity is given by \Delta v = at.

Acceleration \vec a is the rate of change of velocity. Average acceleration is given by a = \frac{\Delta v}{t}. The direction of acceleration is equal to that of the force producing the acceleration.

Equations of linear motion

These equations apply only to motion in a straight line with uniform acceleration. v_i is initial velocity, v_f final velocity, and d total displacement.

  • v_f = v_i + at
    This is because at is equal to the change in velocity.
  • v_f^2 = v_i^2 + 2ad
    I don’t understand this one.
  • d = \left( \frac{v_f + v_i}{2} \right) t
    This is because we multiply average velocity by time elapsed.
  • d = v_i t + \frac{1}{2}at^2
    This is an expansion of the previous equation, after replacing v_f with v_i + at as per the first equation. It makes sense geometrically, where graphing v over t gives a straight line.

Momentum

Momentum (p) is measured in \kg\m\s^{-1} or newton-seconds. It is given by:

\vec{p} = m \vec{v} \tag{mass, velocity}

Impulse

Change in momentum is called the impulse of a force, and is given by:

\Delta \vec{p} = m \Delta \vec{v} = \vec{F} \Delta t

This can be found by rearranging F = m\frac{\Delta v}{\Delta t}.

Conservation of momentum

Momentum is always conserved following an explosion or collision, but kinetic energy is not (it is spent to change velocity). Collisions which involve a loss of energy are called inelastic collisions. Collisions which do not involve a loss of energy are called elastic collisions.

If objects A and B collide, the change in momentum of one is equal and opposite to the change in momentum of the other:

\Delta\vec{p}_A = -\Delta\vec{p}_B

A collision of this type is called a glancing collision.

Conservation of momentum in two dimensions

Find the momentum of each object as a vector, and then find the sum the vectors using a vector diagram.

Subtracting a vector is the same as adding the opposite vector (flip the magnitude or the direction):

\vec{A} - \vec{B} = \vec{A} + -\vec{B}

Center of mass

The center of mass of an object can be found by suspending the object from several points in turn, and finding the intersection between the lines drawn downwards from each point of suspension. This is because the center of mass will always lie directly beneath a point of suspension.

The center of mass of a two particle system is the point that would be the center of mass if both particles were joined by a weightless bar.

Two objects in space will orbit around their common center of mass.

In any situation where momentum is conserved, the velocity of the center of mass remains constant:

\vec{v}_{total} = \frac{\vec{p}_{total}}{m_{total}}

The distance r of the center of mass of n objects on a line from an arbitrary point will be:

r = \frac{\sum_{i=1}^n m_i r_i}{\sum_{i=1}^n m_i}

Circular motion

Circular motion concerns the translational motion of an object that is rotating around a pivot at a constant speed.

Circular velocity

An object travelling in a circle is always changing direction, so its velocity is always changing. The velocity at any point in time is tangent to the circle traced, and if released the object will travel in this direction.

The speed of the object (v = \frac{d}{t}) can be determined from the radius r of the circle traced by the center of mass, and the period T taken to trace a full circle. The distance travelled in one period T is equal to 2\pi r (as per the circumference of a circle), and so:

v_c = \frac{2\pi r}{T}

Centripetal force

The acceleration of an object can be calculated from the velocities of two nearby points \vec{v_i} and \vec{v_f} on the circle (initial and final). Acceleration is given by \vec{a} = \frac{\Delta\vec{v}}{\Delta t}, which gives a vector pointing to the center of the circle. This is because the pivot is applying a centripetal force that pulls the object back towards the center.

a_c = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{v^2}{r}

The centripetal force can be found by merging the definition of a_c with the force equation:

F_c = \frac{mv_c^2}{r}

These can be rewritten further, merging in the formula for v_c:

a_c = \frac{4\pi^2r}{T^2}
F_c = \frac{m \: 4\pi^2r}{T^2}

Satellites

The force of gravity between two free bodies is given by

F_C = \frac{G M_1 M_2}{r^2}

where G is the universal gravitational constant (6.67 \times 10^{-11}), M_1 and M_2 are the masses involved, and r is the distance between them. This force is equal to the centripetal force required to keep one of the bodies in orbit with the other.

\frac{G M_1 M_2}{r^2} = \frac{M_2 v^2}{r}

Cancel and rearrange to make v the subject. The speed of a satellite in an orbit of radius r, about a planet of mass M, is given by

v_c = \sqrt{\frac{G M}{r}}

Note that this formula does not include the mass of the satellite. All satellites, no matter the mass, have the same orbital speed for a given radius.

Conical pendulums

A conical pendulum swings around in a circle instead of back and forth, so that the string traces a cone shape. The only two forces acting on the mass are the weight force F_W (dead downwards) and the tension force T (diagonally upwards along the string).

The angle from vertical \theta of the string is measured from the vertical axis (the direction of F_W), not the horizontal axis as is common in math. The tension force can be decomposed into a vertical component T\cos\theta and a horizontal component T\sin\theta (this makes sense if you draw the force diagram, the vertical axis is the adjacent and the horizontal is the opposite).

Since the mass m is not moving in the vertical direction (the pendulum traces a circle on the horizontal plane), the two vertical forces must be equal and opposite. The resultant force on m is just the horizontal component. Since this force acts towards the center of the circle, it acts as a centripetal force and causes m to move with circular motion.

This gives us everything we need to take a mass m, an angle from vertical \theta, a radius r, and find the centripetal force and the speed of the mass.

Curved roads

Driving a vehicle around a circular track is another example of circular motion. The vehicle must provide sufficient centripetal force (through friction or a reaction force) to traverse the curve without sliding. The reaction force from a surface can only act perpendicular to the surface.

We are able to determine the maximum speed that a mass m is able to traverse a circle of radius r given a maximum friction force F.