PHYSICS PART 1 TEXTBOOK FOR CLASS XI>Chapter -1 - Systems of Particles and Rotational Motion>EXERCISES>Q 7.24

MEDIUM

11th CBSE

IMPORTANT

Earn 100

A bullet of mass $10 g$ and speed $500 m / s$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0 m$ wide and weighs $12 kg$ . It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it. (Hint: The moment of inertia of the door about the vertical axis at one end is $\frac{M L^{2}}{3}$ )

Important Points to Remember in Chapter -1 - Systems of Particles and Rotational Motion from NCERT PHYSICS PART 1 TEXTBOOK FOR CLASS XI Solutions

1. Centre of mass of a system of n discrete particles:

${\vec{r}}_{c m} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2} + \dots \dots \dots + m_{n} {\vec{r}}_{n}}{m_{1} + m_{2} + \dots \dots . . + m_{n}}$

2. Centre of mass of a continuous mass distribution:

$x_{c m} = \frac{\int x d m}{\int d m}, y_{c m} = \frac{\int y d m}{\int d m}, z_{c m} = \frac{\int z d m}{\int d m}$

3. Velocity of centre of mass of system:

(i) ${\vec{v}}_{c m} = \frac{m_{1} {\vec{v}}_{1} + m_{2} {\vec{v}}_{2} + \dots \dots \dots + m_{n} {\vec{v}}_{n}}{M}$

(ii) Momentum of a system given by, ${\vec{P}}_{S y s t e m} = M {\vec{v}}_{c m}$

4. Acceleration of centre of mass of system:

(i) ${\vec{a}}_{c m} = \frac{m_{1} {\vec{a}}_{1} + m_{2} {\vec{a}}_{2} + \dots \dots \dots + m_{n} {\vec{a}}_{n}}{M}$
(ii) The net force acting on a system is given by, ${\vec{F}}_{e x t} = M {\vec{a}}_{c m}$

5. Cross product:

(i) Cross product of two vectors $\vec{A}$ and $\vec{B}$ can be written as $\vec{A} \times \vec{B} = A B \sin θ \hat{n}$ , where $θ$ is the angle between the vectors, $\hat{n}$ is unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ .

(ii) If $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$ and $\vec{B} = B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}$ then, $\vec{A} \times \vec{B} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{matrix}|$

6. Moment of Inertia:

(i) For a single particle, $I = m r^{2}$ where $m$ is the mass of the particle and $r$ is the perpendicular distance of the particle from the axis about which moment of Inertia is to be calculated.

(ii) Moment of inertia of a system of particles: $I = m r_{1}^{2} + m_{2} r_{2}^{2} + . . .$ $= I_{1} + I_{2} + I_{3} + . . . .$

(iii) Moment of inertia of a continuous object: $I = \int d m r^{2}$ , where $d m$ is the mass of a small element, $r$ is the perpendicular distance of the element from the axis

7. Perpendicular Axis Theorem (Only applicable to plane lamina):

$I_{z} = I_{x} + I_{y}$ (when object is in $x - y$ plane).

8. Parallel Axis Theorem (Applicable to any type of object):

$I = I_{c m} + M d^{2}$ , here $I_{c m}$ is the moment of inertia about an axis passing through centre of mass and parallel to required axis and $d$ is the perpendicular distance between required axis and axis passing through centre of mass.

9. Moment of inertia of few structures:

(i) Solid sphere: $\frac{2}{5} M R^{2}$

(ii) Hollow sphere: $\frac{2}{3} M R^{2}$

(iii) Ring: $M R^{2}$

(iv) Disc: $\frac{1}{2} M R^{2}$

(v) Hollow cylinder: $M R^{2}$

(vi) Solid cylinder: $\frac{1}{2} M R^{2}$

(vii) Rod about centre: $\frac{M L^{2}}{12}$

(viii) Rectangular plate: $\frac{M (a^{2} + b^{2})}{12}$

10. Moment of inertia ( $I$ ) in terms of radius of gyration $K$ :

$I = M K^{2}$

11. Torque:

(i) $\vec{τ} = \vec{r} \times \vec{F}$

(ii) $|τ| = r F \sin θ = r_{⊥} F = F_{⊥} r$

12. Angular momentum $(\vec{L})$ of a particle about a point:

(i) $\vec{L} = \vec{r} \times \vec{P}$

(ii) $|L| = r P \sin θ = r_{⊥} P = P_{⊥} r$

13. Angular momentum of a rigid body:

(i) $L_{H} = I_{H} ω$ , where $L_{H} =$ angular momentum of object about axis $H$ . $I_{H} =$ Moment of Inertia of rigid object about axis $H$ . $ω =$ angular velocity of the object.

(ii) ${\vec{L}}_{A B} = I_{c m} \vec{ω} + {\vec{r}}_{c m} \times (M {\vec{v}}_{c m})$

14. Conservation of angular momentum:

According to conservation of angular momentum, if $τ_{e x t} = 0$ about a point or axis of rotation, angular momentum of the particle or the system remains constant about that axis or point.

15. Relation between Torque and Angular Momentum:

$\vec{τ} = \frac{d \vec{L}}{d t}$

16. Kinetic energy:

Total K.E. $= \frac{1}{2} M v_{c m}^{2} + \frac{1}{2} I_{c m} ω^{2}$