Motivation

The motion of point particles.

If one has extended objects like a football, in kinematics, football is represented by a point particle.

Ideal rigid body is one which does not deform, and shape does not change.

1. Translational motion

Slipping

Skidding

Wheel rotating about an axis and moving, has got both rotational motion and translational motion.

All the particles of body have the same velocity $\vec{v}$: Translational motion.

The instantaneous velocity is zero, so different points have different velocities.

Rotational motion: rotation of rigid body about a fixed axis.

Different points on rigid body, having different linear velocities.

The ball goes without rotating about any axis: Pure translation

Rotation about a fixed axis: Rotational motion

Applying a normal force or some force which is making an angle to the door.

If one apply the force at the hinges, the door is not going to rotate.

Rotation is all about the point of application of a force.

Standard example: Pedestal fan

Blade rotates and you get the air.

$X_{C M}=\frac{m_1 x_1+m_2 x_2}{m_1+m_2}$

if $m_1 = m_2$

$X_{C M}=\frac{x_{1+} x_2}{2}$

Several Particles System

$X_{c_M}=\frac{m_1 x_1+m_2 x_2+\cdot +m_n x_n}{\left(m_1+m_2+\cdot+m_n\right)}$

$X_{C M}=\frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}$

$\vec{r}=x \hat{\imath}+y \hat{\jmath}+z\hat{k}$

The unit vectors, $\hat{e}_x, \hat{e}_y \operatorname{and} \hat{e}_z$

$\vec{r}=x \hat{e}_x+y \hat{e}_y+z \hat{e}_z$

$\vec{r}_{cm}$= $\frac{\sum m_1 \vec{r}_1}{\sum m_1}$

$X_{C M}=\frac{\sum_{i=1}^N\left(\Delta m_i\right) x_i}{\sum_{i=1}^N\left(\Delta m_i\right)} \rightarrow \frac{\int x d m}{\int d m}$

$\vec{r}_{C M}=\frac{\int \vec{r} d m}{\int d m}$

Choose the center of sun as the origin.

$m_1$ = Mass of the Sun = $2 \times 10^{30}$ kg

$m_2$ = Mass of the Earth = $6 \times 10^{24}$ kg

$r_1$ = Distance of the Sun from the origin = 0 (since the origin is the center of the Sun)

$r_2$ = Distance of the Earth from the Sun = $1.5 \times 10^{11}$ meters

$R = \frac{m_1 \times r_1 + m_2 \times r_2}{m_1 + m_2}$

$R = \frac{(2 \times 10^{30} \times 0) + (6 \times 10^{24} \times 1.5 \times 10^{11})}{2 \times 10^{30} + 6 \times 10^{24}}$

$R \approx 4.5 \times 10^{5} \text{ meters}$

$(-1,1) =-1\hat{i}+1 \hat{j} =-\hat{i}+\hat{j}$

$\vec{R} =\frac{1(-1,1)+2(1,1)+1(1,-1)+2(-1,-1)}{6}$

$=\frac{(0,0)}{6}=(0,0)$

$\vec{R}=(0,0)$