Center of Mass JEE 2024 PDF

# Arjuna JEE 2024 ## Com - 01 - Subject Name - Physics - Chapter Name - - Lecture No. - 01 - By - Saleem Ahmed Sir ## Today's Goal 1. Com Introduction 2. Location of Com 3. 4. ## Com ![Com description] - A diagram with three examples of rigid bodies: - **Pure Translation:** A vertical bar wi...

# Arjuna JEE 2024 ## Com - 01 - Subject Name - Physics - Chapter Name - - Lecture No. - 01 - By - Saleem Ahmed Sir ## Today's Goal 1. Com Introduction 2. Location of Com 3. 4. ## Com ![Com description] - A diagram with three examples of rigid bodies: - **Pure Translation:** A vertical bar with four arrows pointing to the right, representing the velocity of each point along the bar. - **Purely Rotational Motion:** A vertical bar with a single arrow pointing along the top of the bar representing the angular velocity ω. The center of rotation is a point at the center of the bar. - **Combined Rotational and Translation Motion:** A vertical bar is moving to the right with a velocity v and rotating around a point at its bottom with angular velocity ω. - Pure Translation motion - Purely Rotational motion (Fix Axis Rotational motion) - CRTM (Combined Rotational + Translation motion) ## Location of com ### For discrete particle system - The x coordinate of the center of mass is defined as: $x_{com} = \frac{m_1x_1 + m_2x_2 + m_3x_3 + ...}{m_1 + m_2 + m_3 + ...} = \frac{\sum_{i=1}^n m_ix_i}{\sum_{i=1}^n m_i}$. - The y coordinate of the center of mass is defined as: $y_{com} = \frac{m_1y_1 + m_2y_2 + m_3y_3 + ...}{m_1 + m_2 + m_3 + ...} = \frac{\sum_{i=1}^n m_iy_i}{\sum_{i=1}^n m_i}$. - The z coordinate of the center of mass is defined as: $z_{com} = \frac{m_1z_1 + m_2z_2 + m_3z_3 + ...}{m_1 + m_2 + m_3 + ...} = \frac{\sum_{i=1}^n m_iz_i}{\sum_{i=1}^n m_i}$. ![Location of com] - Graphical representation of 3 particles in a 2D coordinate system. The x, y, and z coordinates of each particle are denoted. #### Find Location of Com * **Scenario:** Three particles with masses 2kg, 3kg, and 5kg, positioned at (3, 1), (4, 7), and (10, 1) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{2 \times 3 + 3 \times 4 + 5 \times 10}{2 + 3 + 5} = \frac{68}{10}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{2 \times 4 + 3 \times 7 + 5 \times 1}{2 + 3 + 5} = \frac{8 + 21 + 5}{10} = \frac{34}{10}$. - Therefore, the center of mass is located at: $Com = (\frac{68}{10},\frac{34}{10})$. * **Scenario:** Four particles with masses 10kg, 5kg, and 3kg located at (-1, 2), (2, 3), (10, -2) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{10 \times (-1) + 5 \times 2 + 3\times 10}{10 + 5 + 3} = \frac{-10 + 10 + 30}{10+5+3} = \frac{30}{18}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{10*2 + 5*3 + 3*(-2)}{10 + 5 + 3} = \frac{20 + 15 - 6}{18} = \frac{29}{18}$. - Therefore, the center of mass is located at: $Com = (\frac{30}{18},\frac{29}{18})$. * **Scenario:** Four particles with masses 3kg, 4kg, and 2kg, positioned at (2, 5, 4), (4, 3, 2), (3, -2, -6) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{6 + 16 + 6}{3 + 4 + 2} = \frac{28}{9}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{15 + 12 + (-4)}{9} = \frac{23}{9}$. - The z-coordinate of the center of mass is: $z_{com} = \frac{12 + 8 -12}{9} = \frac{8}{9}$. * **Scenario:** Five particles with masses, 4kg, 3kg, 2kg, and 1kg located at (3, 4), (5, 0), (0, 10), and (-4, -4) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{0 + 12 + 15 - 4}{10} = \frac{23}{10}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{20 + 16 + 0 -4}{10} = \frac{32}{10}$. * **Scenario:** Three particles with masses 1kg, 2kg, and 3kg located at (5, 5√3), (0, 0), and (10, 0) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{0 + 5 + 30}{6} = \frac{35}{6}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{0 + 5√3 + 0}{6} = \frac{5√3}{6}$. * **Scenario:** Two particles with masses 4kg and 2kg located at (2, 3) and (3, 2) respectively. - The x-coordinate of the center of mass is: $x_{com} = \frac{8 + 6}{4 + 2} = \frac{14}{6}$. - The y-coordinate of the center of mass is: $y_{com} = \frac{12 + 4}{6} = \frac{16}{6}$. - Position vector of com of system: $\frac{14}{6}\hat{i} + \frac{16}{6}\hat{j}$ ### Position Vector of Com - $r_{com} = \frac{m_1\vec{r_1} + m_2\vec{r_2}}{m_1 + m_2}$. - The position vector of the center of mass is: $ \vec{r_{com}} = 4(2\hat{i} + 3\hat{j}) + 2(3\hat{i} + 2\hat{j}) = \frac{8\hat{i} + 12\hat{j} + 6\hat{i} + 4\hat{j}}{4 + 2} = \frac{14\hat{i} + 16\hat{j}}{6} = \frac{14}{6}\hat{i} + \frac{16}{6}\hat{j}$. ### System of Discrete Particle - $\vec{r_{com}} = \frac{m_1\vec{r_1} + m_2\vec{r_2} + m_3\vec{r_3}}{m_1 + m_2 + m_3 ...}$. ![System of discrete particles] - A diagram with 4 particles. The origin is at the center of the diagram. The position vectors of each particle are represented with arrows pointing from the origin to each of the particle locations. #### Example * **Scenario:** Two particles with masses 2kg and 4kg. - The x-coordinate of the center of mass is: $x_{com}= \frac{0 + 24}{2 + 4} = 4$. - The y-coordinate of the center of mass is: $y_{com}=0$. - Therefore, $com = (4, 0)$. ![Example scenario] - A graphic representation of 2 particles with mass 2kg and 4kg positioned on the x-axis with their center of mass, com, at (4,0). ## Additional Information - **Scenario:** With two particles, the center of mass will lie on the line joining the two particles and will be closer to the heavier particle. - **Geometric shapes:** The center of mass of a uniform rod lies at the center of the rod, the center of mass of a uniform disc or sphere lies at the center of the disc or sphere, a uniform ring lies at the center of the ring, a rectangular sheet lies at the center of the rectangle. ![Geometric Shapes] - A diagram with five geometric objects showing the center of mass (com) for: - uniform rod - uniform disc - ring (uniform) - rectangular sheet - square plate (uniform) ## Home Work - Enjoy your Sunday ## Thank You