Newton Laws Of Motion
Third Law Of Motion:
$$\vec{F} _{AB} = \vec{F} _{BA} $$
$$ \vec{\mathrm{F}}_{\mathrm{AB}}=\text { Force on } \mathrm{A} \text { due to } \mathrm{B} $$
$$\vec{\mathrm{F}}_{\mathrm{BA}}=\text { Force on } \mathrm{B} \text { due to } \mathrm{A}$$
From Second Law Of Motion:
PYQ2023LawsOfMotionQ1, PYQ2023LawsOfMotionQ3, PYQ2023LawsOfMotionQ4, PYQ2023LawsOfMotionQ5, PYQ2023LawsOfMotionQ7, PYQ2023LawsOfMotionQ8, PYQ2023LawsOfMotionQ10, PYQ2023LawsOfMotionQ13, PYQ2023LawsOfMotionQ15
$$F_x =\frac{dP_x}{dt}=ma_x \quad F_y =\frac{dP_y}{dt}=ma_y \quad F_y =\frac{dP_y}{dt}=ma_y $$
Weighing Machine :
A weighing machine does not measure the weight but measures the force exerted by object on its upper surface.
Spring Force:
$$\vec{F}=k \vec{x}$$
x is displacement of the free end from its natural length or deformation of the spring where $\mathrm{k}=$ spring constant.
Spring Property:
$\mathrm{K} \times \ell=$ constant = Natural length of spring.
 If spring is cut into two in the ratio $m: n$ then spring constant is given by
$$l_1 = \frac{ml}{m+n}; \quad l_2 = \frac{nl}{m+n}; \quad k_l = k_1 l_1 = k_2 l_2$$
 For series combination of springs
$$\frac{1}{k_{e q}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}+\ldots$$
 For parallel combination of spring
$$k_{\text {eq }}=k_{1}+k_{2}+k_{3} \ldots$$
Spring Balance:
It does not measure the weight. It measures the force exerted by the object at the hook.
Remember :
$$a = \frac{(m_2  m_1)g}{m_1 + m_2}$$
$$T = \frac{2m_1m_2g}{m_1+m_2}$$
Wedge Constraint :
Components of velocity along perpendicular direction to the contact plane of the two objects is always equal if there is no deformations and they remain in contact.
Newton’s Law For A System:
$$\sum \vec{F} _\text{ext} = m \cdot \vec{a}$$ where: $\vec{a}$ is the acceleration of the entire system. $sum \vec{F} _\text{ext}$ is the vector sum of all external forces acting on the system.
Newton’s Law For Non Inertial Frame :
In a noninertial frame, Newton’s second law for a system of objects is given by:
$$\sum \vec{F} _\text{total} = \sum \vec{F} _\text{external} + \sum \vec{F} _\text{pseudo} = m\vec{a}$$
Where:

$\sum \vec{F}_\text{total}$ is the total force in the noninertial frame.

$\sum \vec{F}_\text{external}$ is the sum of external forces.

$\sum \vec{F}_\text{pseudo}$ is the sum of pseudoforces introduced due to the frame’s acceleration.

$\vec{a}$ is the acceleration of the entire system.
(a) Inertial reference frame: Frame of reference moving with constant velocity.
(b) Noninertial reference frame: A frame of reference moving with nonzero acceleration.
Friction
PYQ2023LawsOfMotionQ13, PYQ2023RotationalMotionQ7
Friction force is of two types:
(a) Kinetic
(b) Static
Kinetic Friction:
$$f_{k}=\mu_{k} N$$
The proportionality constant $\mu_{\mathrm{k}}$ is called the coefficient of kinetic friction and its value depends on the nature of the two surfaces in contact.
Static Friction:
It exists between the two surfaces when there is tendency of relative motion but no relative motion along the two contact surfaces.
This means static friction is a variable and self adjusting force. However it has a maximum value called limiting friction.
$$f_{\max }=\mu_{s} N $$
$$ 0 \leq f_{s} \leq f_{s \max }$$
Elevator Problem:
 Body is inside an elevator (lift) that is moving upward: $$W_{\text{apparent}} = m \cdot (g + a)$$
Where: a is the acceleration of the elevator.
 When an elevator (lift) is moving downward: $$W_{\text{apparent}} = m \cdot (g + a)$$
Centripetal Force:
PYQ2023LawsOfMotionQ9, PYQ2023LawsOfMotionQ12, PYQ2023GravitationQ9
The centripetal force $(F_c)$ formula is given by:
$$F_c = \frac{{m \cdot v^2}}{{r}}$$
Where:

$F_c$ is the centripetal force (in newtons, N).

$m$ is the mass of the object in circular motion (in kilograms, kg).

$v$ is the velocity of the object in circular motion (in meters per second, m/s).

$r$ is the radius of the circular path (in meters, m).