Quantities
A scalar quantity has a magnitude, but no direction. A scalar is notated as a plain letter, as s.
A vector quantity has both a magnitude and a direction (in any number of dimensions). A vector is notated as a letter with an arrow above, as \vec{v}.
Force
Force (F) is measured in newtons (N), equivalent to \kg\m\s^{-2}.
Weight force is force due to gravity:
Energy
Energy (E) is measured in joules (J), equivalent to newton-meters , the same as energy. or \kg\m^2\s^{-2}. It is force over distance.
Kinetic energy (E_k) is energy posessed from speed (not velocity), or energy required to attain a speed, and is measured in joules. It is given by:
Gravitational potential energy (E_p) is energy posessed from height, or energy required to attain a height (or oppose gravity over a distance), and is measured in joules. It is given by:
Work
Work (W) is done when energy is transformed from one form to another (measured in joules).
The work done when using a force F to move an object by a distance d is given by:
For a force that varies over time, the work done is given by the total area beneath a graph of force vs distance, which can be calculated via definite integration.
Power
Power (P) is measured in watts (W). It is the rate of change of work, measuring the rate in which work is done, and is given by:
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:
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
Position is given as either:
- A scalar d (distance travelled), the distance between two points.
- A vector \vec{d} (displacement), the distance and direction of a point from a starting point. This can be different from distance if the direction of motion changes in-between points.
Rate of change of position is given as either:
- A scalar v (speed), the distance travelled in a unit of time (v = \frac{\Delta d}{t}).
- A vector \vec{v} (velocity), the distance and direction travelled in a unit of time.
Second-degree rate of change of position is given as:
- A vector \vec{a} only (acceleration), the change in velocity in a unit of time (a = \frac{\Delta v}{t}). The direction of acceleration is equal to the direction of the force producing the acceleration. Notice that we can only find the change in velocity, not the velocity outright.
To recap:
- a change in velocity (\Delta v) is calculated from a constant acceleration applied over a duration (at).
- an average acceleration (a) is calculated from a change in velocity over a duration (\frac{\Delta v}{t}).
- a change in distance (\Delta d) is calculated from a constant velocity applied over a duration (vt).
- an average velocity (v) is calculated from a change in distance over a duration (\frac{\Delta d}{t}).
At this point, we’re only concerned with constant acceleration and linear velocity.
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:
Impulse
Change in momentum is called the impulse of a force, and is given by:
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:
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):
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:
The distance r of the center of mass of n objects on a line from an arbitrary point will be:
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:
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.
The centripetal force can be found by merging the definition of a_c with the force equation:
These can be rewritten further, merging in the formula for v_c:
Satellites
The force of gravity between two free bodies is given by
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.
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
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.
Rotation and torque
Pure translation is movement of the center of mass of an object without any rotation. This is caused by a force acting through the position of the center of mass.
Pure rotation is circular movement about the center of mass while the center of mass remains stationary. This is caused by a force couple, where two equal and opposite forces act apart on the object (often an applied force combined with a reaction force from an axle or pivot).
A single force that does not pass through the center of mass will impart both translation and rotation.
Rotational units
Angular displacment \theta is a measure of change of rotation, measured in \rad.
Angular velocity \omega is a measure of rate of change of rotation, measured in \rad \s^{-1}.
The angular velocity \omega and the translational velocity v at a point distance r from the pivot are related by:
Angular acceleration \alpha is a measure of second degree rate of change of rotation, measured in \rad \s^{-2}.
The angular velocity \alpha and the translational velocity a at a point distance r from the pivot are related by:
Equations of rotational motion
These equations apply only to rotational motion with uniform angular acceleration. \omega_i is initial angular velocity, \omega_f final angular velocity, and \theta total displacement.
- \omega_f = \omega_i + \alpha t
- \omega_f^2 = \omega_i^2 + 2\alpha\theta
- \theta = \left( \frac{\omega_f + \omega_i}{2} \right) t
- \theta = \omega_i t + \frac{1}{2}\alpha t^2
These are exact analogues of the equations of linear motion above.
Torque
The turning effect of a force is called torque (\tau). Torque is proportional to force applied and distance from the pivot, and is measured in newton-meters (equivalent to joules).
When a force is applied to a wheel, a reaction force is produced by the axle. Since this reaction force acts through the center of rotation, it does not produce any torque, so the turning effect is calculated only from the applied force and the distance from the pivot.
Torque is directly proportional to angular acceleration: the greater the torque, the greater the acceleration. The constant I is called the rotational inertia.
Rotational inertia
Rotational interia is measured in \kg \m^2, and depends on the mass of the object and the distance of the mass from the center of rotation. It’s analogous to mass m in the force equation, because an object with a larger mass will be harder to set spinning.
If opposing masses were placed on a spinning disk, the rotational inertia would be greater the further towards the edge the masses were placed, even though the center of mass remains the same.