Is it necessary that a mass should be present at the centre of mass of a... | Physics

SYSTEM OF PARTICLES AND ROTATIONAL MOTION

Very Short Question Answers

Q1. Is it necessary that a mass should be present at the centre of mass of any system?
Ans.

Not necessary, Ex: Uniform ring

Q2. What is the difference in the positions of a girl carrying a bag in o e of her hands and another girl carrying a bag in each of her two hands?
Ans.

In the case of first girl who is carrying a bag in one hand, centre of mass. Shift towards
the hand in which there is bag.
In the case of second girl, the position of the centre of mass does not change.

Q3. Two rigid bodies have same moment of inertia about their axes of symmetry. Of the two,which body will have greater kinetic energy?
Ans.

$https://latex.codecogs.com/png.image?\inline \large \dpi{110} Kinetic\: Energy \: K=\frac{L^{2}}{2I}$

Q4. Why are spokes provided in a bicycle wheel?
Ans.

To increase the moment of inertia

Q5. We cannot open or close the door by applying force at the hinges. Why?
Ans.

Torque τ = 𝑟𝐹 sin 𝜃
If you are applying force at the hinges means 𝑟 = 0
Then torque τ = 0, there is no turning effect.

we cannot open or close the door by applying force at the hinges.

Q6. Why do we prefer a spanner of longer arm as compared to the spanner of shorter arm?
Ans.

Torque τ = 𝑟𝐹 sin 𝜃
In the case of a spanner of longer arm, length of the lever of force i.e. ‘𝑟’ is more, so torque
(τ) is more. Hence turning effect is more.
∴ spanner of longer arm is preferred.

Q7. By spinning eggs on a table top, how will you distinguish a hardboiled egg from a raw egg?
Ans.

Raw egg stops sooner than boiled egg.

Q8. Why should a helicopter necessarily have two propellers?
Ans.

If there were one propeller, then according to law of conservation of angular momentum,the helicopter would rotate itself in the opposite direction.
∴ Helicopter are provided with two propellers.

Q9. If the polar ice caps of the ea th were to melt, what would the effect of the length of the day be?
Ans.

If the polar ice melt, the water formed will spread over the surface of the earth. So, the moment of inertia (𝐼) will increase. According to law of conservation of angular momentum,(Iω=constant) angular velocity (ω) of the earth will decrease. Hence the length of the day will
increase (∴ 𝑇 = ω/2π)

Q10. Why is it easier to balance a bicycle in motion?
Ans.

The rotating wheels of a bicycle possess angular momentum. According to law of conservation of angular momentum, the direction of the angular
momentum is along the axis of wheel. So, the bicycle does not get tilted.

Short Question Answers

Q1. Distinguish between centre of mass and centre of gravity?
Ans.

Center of mass	Center of gravity
1. It is a point at which total mass appear to be concentrated.	1. It is a point where the total weight of the body acts
2. It refers mass of the body	2. It refers to the weight of the body
3.It depends on mass distribution	3. It depends on acceleration due to gravity
4. It may or may not lie inside the body	4.It always lies inside of the body.

Q2. Show that a system of particles moves under the influence of an external force as if the force is applied at its centre f mass?

Ans. Consider a system particles of mass $m_1,m_2,m_3,....., m_n$ and their positions $r_{1},r_{2},r_{3},.....,r_{n}$ Therefore,the centre of mass of the system is $R_{cm}$
$$\begin{gathered}
R_{cm}={\frac{m_{1}r_{1}+m_{2}r_{2}+m_{3}r_{3}+\cdots+m_{n}r_{n}}{m_{1}+m_{2}+m_{3}+\cdots+m_{n}}} \\
R_{cm}=\frac{m_{1}r_{1}+m_{2}r_{2}+m_{3}r_{3}+\cdots+m_{n}r_{n}}{M} \\
MR_{cm}=\:m_{1}r_{1}+m_{2}r_{2}+m_{3}r_{3}+\cdots+m_{n}r_{n} \\
MV_{cm}=\:m_{1}v_{1}+m_{2}v_{2}+m_{3}v_{3}+\cdots+m_{n}v_{n} \\
Ma_{cm}=\:m_{1}a_{1}+m_{2}a_{2}+m_{3}a_{3}+\cdots+m_{n}a_{n}
\end{gathered}$$Therefore

$$F_{ext}=\:f_{1}+f_{2}+f_{3}+\cdots+f_{n}$$

Hence , the system of particles moves under the influence of external force as if the forces is applied at its centre of mass.

Q3. Explain about the Centre of mass of Earth - moon system and its rotation around the sun?

Ans. The mass ofthe earth isgreater 81 times that ofthe moon mass.Thus,the centre of mass of the Earth-moon system is located relatively near to the centre of the earth. The Gravitational attraction of. the sun is an external force on the Earth -moon system. Hence, the centre of mass Earth-moon system. moves in elliptical path around the sun.

Q4. Define vector product .Explain the properties of vector product with two examples?

Ans. Vector Product: The cross product of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ defined as follows

$$\overrightarrow{A}\times\overrightarrow{B}=ab\sin\theta\:\widehat{n}$$

Where $\theta$ - angle between two vectors

$\hat{n}$ - perpendicular unit vector to$ A,B$ plane

Properties: 1. It is does not obey Commutativelaw.Itis anti-commutative $\vec{A}\times\vec{B}=-\vec{B}\times\vec{A}$

2.It is obey distributive law $\vec{A}\times\left(\vec{B}+\vec{C}\right)=\left(\vec{A}\times\vec{B}\right)+\left(\vec{A}\times\vec{C}\right)$

3. Between two parallel vectors $\overrightarrow{A} \times {\overrightarrow{B}}=\mathbf{0}$ $$ i\times i=j\times j=k\times k=0$$

4.Between two perpendicular vectors $ \overrightarrow{A}\times\overrightarrow{B}=ab\:\widehat{n}$

Examples: 1. Angular momentum $\vec{L} =$ $\vec{r} \times$ $\vec{P}$

2. Torque $\vec{\tau}=\vec{r}\times\vec{F}$

Q5. Define angular velocity . Derive$ v =r\omega$ ?

Ans. Angular velocity $(\omega)$ : The rate of change of angular displacement is in known as angular velocity

$$\omega=\frac{d\theta}{dt}$$

SI Unit: rad/s

Derivation $\underline {V=r\omega:}$As shown in the diagram Let consider a particle moving along the circular patlh and Let $r$ is radius of the circle, $ x $ is arc length, $ \theta $ is angular displacement $ V $ is linear velocity and $\omega $ is angular velocity$$x=r\theta $$Differentiating above equation with respect to time $t$. $$\begin{aligned}
&{\frac{dx}{dt}}={\frac{d(r\theta)}{dt}} \\
&V=r{\frac{d(\theta)}{dt}} \\
&V=r\omega \\
\end{aligned}$$

Q6. Define angular acceleration and torque. Establish the relation between angular acceleration and torque?

Ans. Angular acceleration: The rate of change of angular velocity is known as angular acceleration..
$$\alpha=\frac{d\omega}{dt}$$ SI Unit: $rad/s^2$

Torque: It is the product of force and perpendicular distance of the applied force from the axis of ration.
$$\begin{aligned}&\tau=r\times F\\&\tau=r\times ma \:\:\:(\because F=m\alpha)\\&\tau=r\times mr\alpha\:\:\:(\because a=r\alpha)\\&\tau=mr^{2}\times\alpha\\&\therefore\tau=I\alpha\\\end{aligned}$$

Q7. Derive expressions for the final velocity and total energy of a body rolling with out slipping?

Ans. Kinetic Energy of a body without slipping: Let us consider a rigid body rolling on a inclined rough. plane without slipping.Thus,the total kineticenergy is equal to sum of the translation kinetic energy and rotational kinetic energy.

$$\begin{gathered}K.E=(K.E)_{trans}+(K.E)_{rota}\\
K.E=\frac{1}{2}mV_{cm}^{2}+\frac{1}{2}I\omega^{2} \\
where\:I=mK^{2}\:and\:\omega=\frac{V_{cm}}{R} \\
K.E=\frac{1}{2}mV_{cm}^{2}+\frac{1}{2}mK^{2}\left(\frac{V_{cm}}{R}\right)^{2} \\
K.E=\frac{1}{2}mV_{cm}^{2}\left(1+\left(\frac{K}{R}\right)^{2}\right)
\end{gathered}$$

The above equation is represent the total kinetic energy of body Final Velocity.The body is rolling downward on the inclined rough plane from height $h$

$$\begin{gathered}\text{Kinetic Energy= Potential Energy} \\ \frac{1}{2}mV_{cm}^{2}\left(1+\left(\frac{K}{R}\right)^{2}\right)=mgh\\ V_{cm}=\sqrt{\frac{2gh}{\left(1+\left(\frac{K}{R}\right)^{2}\right)}}=\sqrt{\frac{2gl\sin\theta}{\left(1+\left(\frac{K}{R}\right)^{2}\right)}}\end{gathered}$$

Essay Question Answers

Q1. a) State And Prove Parallel Axes Theorem? b) For a thin flat circular disk, the radius of gyration about a diameter as axis is k. If the disk is cut along a diameter AB as shown in to equal pieces, then find the radius of gyration of each piece about AB?