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:

$$F_x =\frac{dP_x}{dt}=ma_x \quad F_y =\frac{dP_y}{dt}=ma_y \quad F_y =\frac{dP_y}{dt}=ma_y $$

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 non-inertial 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 non-inertial frame.

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

• $\sum \vec{F}_\text{pseudo}$ is the sum of pseudo-forces 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) Non-inertial reference frame: A frame of reference moving with non-zero acceleration.

Friction

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:

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).