Physics Lens

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[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

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[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 = $$\Delta p =\int {F .dt}$$

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[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
  • $$u_2-u_1=v_1-v_2$$

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[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.

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[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

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[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.

• 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:

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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.

• 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.

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: $a^2 = b^2 + c^2-2bc \cos \alpha$