Types of Forces

• Static friction
  • Frictional force between surfaces at rest with respect to each other.
  • It increases with increasing applied force up to a maximum value (which is proportional to normal contact force).
• Kinetic friction
  • Frictional force acting between surfaces in relative motion.
• Viscous forces
  • Resistive force experienced by a solid moving in a fluid.
  • Dependent on speed of object v, e.g. $$F_D\propto v$$ at low speeds and $$F_D\propto v^2$$ at high speeds.

[accordions autoHeight='true'] [accordion title="1. Newton's Laws of Motion"]

• Newton's First Law:  a body will remain in its state of rest or uniform motion in a straight line unless acted upon by a resultant force.
• Newton's Second Law:  the rate of change of momentum of a body is proportional to the resultant force acting on it and the change takes place in the direction of the resultant force.
  • $$F =\frac{dp}{dt}$$ in general
  • $$F =ma$$ when mass is constant.
• Newton's Third Law:  if body A exerts a force on body B, then body B exerts an equal and opposite force on body A

[/accordion] [accordion title="2. Linear Momentum"]

• The linear momentum of a body is defined as the product of its mass and its velocity.
• Impulse is the product of the force acting on a body and the time interval during which the force is exerted. It is equal to the change in momentum of the body.
  • For constant force, impulse = $$\Delta p =F \Delta t$$
  • In general, impulse = $$\Delta p =\int {F .dt}$$

[/accordion] [accordion title="3. Collision Problems"]

• The principle of conservation of momentum states that the total momentum of a system of colliding objects remains constant provided no resultant external force acts on the system.
• Conservation of momentum applies to both elastic and inelastic collisions.
  • $$m_1u_1+m_2u_2=m_1v_1+m_2v_2$$
• Conservation of kinetic energy applies only to elastic collisions.
  • $$\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2$$
• Relative speed of approach = Relative speed of separation
  • $$u_2-u_1=v_1-v_2$$

[/accordion] [/accordions]

[accordions autoHeight='true'] [accordion title="1. Definitions"]

• Displacement is the distance travelled along a specified direction.
• Speed is the rate of change of distance travelled.
• Velocity is the rate of change of displacement.
• Acceleration is the rate of change of velocity.

[/accordion] [accordion title="2. One-Dimensional Motion with Constant Acceleration"]

• $$v=u+at$$
• $$s=(\frac{u+v}{2})t$$
• $$s=ut+\frac{1}{2}at^2$$
• $$v^2=u^2+2as$$

s: displacement u: initial velocity v: final velocity a: acceleration t: time [/accordion] [accordion title="3. Two-Dimensional Motion"]

• Tip: Sometimes, you will require two equations to solve a kinematics problem. For a parabolic path in a projectile motion without resistive forces, you can draw a table such as the one below and fill in the blank with the information given in the question.

[caption id="attachment_1933" align="aligncenter" width="163"] Initial velocity at an angle[/caption]

• In the case where a projectile is launched at an angle $$\theta$$ to the horizontal and undergoes the acceleration of free fall, the various horizontal and vertical components of displacement, velocity and acceleration can be expressed in the following way:

	Horizontal	Vertical
displacement, s	$$(u \cos \theta)t$$	$$(u \sin \theta)t+\frac{1}{2}gt^2$$
initial velocity, u	$$u \cos \theta$$	$$u \sin \theta$$
initial velocity, v	$$u \cos \theta$$	$$u \sin \theta +gt$$
acceleration, a	0	$$g$$
time, t	same for	both dimensions

[/accordion] [/accordions]

Base and Derived Quantities

• Physical quantities are classified as base (or fundamental) quantities and derived quantities. 7 base quantities are chosen to form the base units.

Base Quantity	Base Unit
mass	kilogram (kg)
length	metre (m)
time	second (s)
electric current	ampere (A)
temperature	kelvin (K)
amount of substance	mole (mol)
luminous intensity	candela (cd)

• Any other physical quantities can be derived from these base quantities. These are called derived quantities.

Prefixes

• Prefixes are attached to a unit when dealing with very large or very small numbers.

Power	Prefix
$10^{-12}$	pico (p)
$10^{-9}$	nano (n)
$10^{-6}$	micro ($\mu$)
$10^{-3}$	milli (m)
$10^{-2}$	centi (c)
$10^{-1}$	deci (d)
$10^3$	kilo (k)
$10^6$	mega (M)
$10^9$	giga (G)
$10^{12}$	tera (T)

Homogeneity of Units in an Equation

• A physical equation is said to be homogeneous if each of the terms, separated by plus, minus, equality or inequality signs has the same base units.

Uncertainty

• Absolute uncertainty of a measurement of $x$ can be written as $\Delta x$. This means that true value of the measurement is likely to lie in the range $x-\Delta x$ to $x + \Delta x$.
• Fractional uncertainty = $\dfrac{\Delta x}{x}$
• Percentage uncertainty = $\dfrac{\Delta x}{x}\times100%$
• If the values of two or more quantities such as $a$ and $b$ are measured and then these are combined to determine another quantity $Y$, the absolute or percentage uncertainty of $Y$ can be calculated as follows:
  • If $Y = a\pm b$, then  $\Delta Y = \Delta a+\Delta b$
  • If $Y = ab$ or $Y = \frac{a}{b}$ , then  $\frac{\Delta Y}{Y} =\frac{\Delta a}{a}+\frac{\Delta b}{b}$
  • If $Y = a^n$ then  $\frac{\Delta Y}{Y} = n\frac{\Delta a}{a}$

Errors

• Systematic errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of same magnitude and sign.
• Random errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of different magnitude and sign.

Accuracy and Precision

• The accuracy of an experiment is a measure of how close a measured value is to the true value. It is a measure of the correctness of the result.
• The precision of an experiment is a measure of how exact the result is without reference to what that the result means. It is a measure of how reproducible the results are, i.e. it is a measure of how small the uncertainty is.

Vectors

• A vector quantity has magnitude and direction.
• A scalar quantity has magnitude only.
• Addition of vectors in 2D: $\vec{a}+\vec{b}=\vec{c}$
• Subtraction of vectors in 2D: $\vec{a}-\vec{b}=\vec{d}$

• Methods of finding magnitudes of vectors:
  1. resolution of vectors into perpendicular components
  2. by scale drawing
  3. using: sine rule: $\frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}$ cosine rule: $a^2 = b^2 + c^2-2bc \cos \alpha$

A disc rotates clockwise about its centre O until point P has moved to point Q, such that OP equals the length of the straight line PQ. What is the angular displacement of OQ relative to OP?

A.   $\frac{\pi}{3}$ rad

B.   $\frac{2\pi}{3}$ rad

C.   $\frac{4\pi}{3}$ rad

D.   $\frac{5\pi}{3}$ rad