Study Material

SYSTEM OF PARTICLES AND ROTATIONAL MOTION

Physics

Inter First Year

English

Very Short Answer Questions

Q1. Explain how a diver jumping into water from a height is able to make a loop in air.

Answer

Ans. According to the law of conservation of angular momentum $I\omega = \text{constant}$, the diver tucks or extends their body to change their moment of inertia ($I$). This causes an inverse change in their angular velocity ($\omega$), allowing them to control their spin and make loops.

Q2. What happens to the duration of a day when polar ice caps melt and flow towards the equator?

Answer

Ans. $I\omega = \text{constant}$, if Earth's polar ice caps melt and flow towards the equator, its moment of inertia ($I$) increases, its angular velocity ($\omega$) must decrease. Therefore, the length of the day increases ($\because T = \frac{2\pi}{\omega}$).

Q3. The stone of a hand flour grinder is provided with a handle near its rim. Give reason.

Answer

Ans. Torque $ \tau = rF\sin\theta$. A handle at the rim produces a greater turning effect.

Q4. By spinning eggs on a table top, how will you distinguish a hard-boiled egg from a raw egg?

Answer

Ans. The raw egg stops sooner than boiled egg. A boiled egg rotates as a rigid body and spins smoothly. A raw egg wobbles and stops sooner because its liquid contents move internally and dissipate energy.

Q5. Why should a helicopter necessarily have two propellers?

Answer

Ans. According to the conservation of angular momentum ($I\omega = \text{constant}$), a helicopter's fuselage tends to rotate opposite to its main propeller. To prevent this, another propeller is provided at the tail end.

Q6. Why is it easier to balance a bicycle in motion?

Answer

Ans. Rotating wheels possess angular momentum. The gyroscopic effect resists sudden changes in orientation and helps stability.

Q7. We cannot open or close the door by applying force at the hinges. Why?

Answer

Ans. Torque $ \tau = rF\sin\theta$. At the hinge r=0, so the applied force produces no turning effect

Q8. A child is sitting at one end of a long trolley moving with a uniform speed $u$, on a smooth horizontal floor. If the child gets up and runs on the trolley in any manner with speed $u$, what will be the Centre of mass of the system (trolley + child)?

Answer

Ans. There is no external horizontal force from the smooth floor ($\vec{F}_{ext} = 0$), the state of the centre of mass remains completely unchanged. The centre of mass of the system will continue to move with the exact same \textbf{uniform speed $u$} in the original direction.

Q9. Newton's law of motion are applicable to the individual particles of a system. However, we can also describe the motion of the system as a whole by Newton's Laws. Justify.

Answer

Ans. According to Newton's third law, all internal forces within the system sum to zero. Therefore, applying Newton's second law to the whole system.
\[ \vec{F}_{ext} = \vec{f}_1 + \vec{f}_2 + \dots + \vec{f}_n = M\vec{a}_{cm} \]
This proves that the entire system behaves exactly as a single particle of mass $M$ located at the centre of mass, moving under the sole influence of external forces.

Q10. In which of the following cases the Centre of mass of a body coincides with its geometric centre? (a) The density continuously decreases from right to left. (b) The density decreases from left to right up to the centre and then increases in the same proportion.

Answer

Ans. (b) The density decreases from left to right up to the centre and then increases in the same proportion.

Q11. Why are spokes provided in a bicycle wheel?

Answer

Ans. To increase the moment of inertia.

Q12. Define radius of gyration. Mention the factors on which it depends.

Answer

Ans. Radius of gyration($k$):The perpendicular distance from the axis of rotation where the entire mass of the body can be concentrated to have the same moment of inertia ($I = Mk^2$).

Radius of gyration $k = \sqrt{\frac{I}{M}}$

Depends on:1. Position and orientation of the axis of rotation. 2. Distribution of mass around the axis. 3.It depends on the shape and size of the body

Q13. A jack screw is provided with a long arm. Explain why?

Answer

Ans. Torque $ \tau = rF\sin\theta$ a long arm produces a greater turning effect.

Q14. A person sits near the edge of a circular platform revolving with a uniform angular speed. What will be the change in the motion of the platform?

Answer

Ans. Moment of inertia increases and angular velocity decreases.

Q15. Mention the factors on which moment of inertia depends.

Answer

Ans. Moment of inertia $I = mr^2$. It is depends on total mass, its distribution, the body’s shape and size.

Q16. Define centripetal force. Write its formula.

Answer

Ans. $F_c=\frac{mv^2}{r}=mr\omega^2$

Centripetal force is the net force that acts perpendicular to a body's velocity, keeping it moving in a circular path. It is always directed inward towards the center of rotation

Q17. It is easier to turn the steering wheel of a larger diameter than that of a small diameter. Give reason.

Answer

Ans. Torque $ \tau = rF\sin\theta$ a larger radius produces a greater turning effect.

Short Answer Questions

Q1. What is meant by Centre of mass of a system? Obtain expression for the Centre of mass of a system consisting of (a) two particles (b) a large number of particles. What is the physical significance of Centre of mass of a system?

Answer

Ans. Centre of mass of a system: It is a point at which total mass appear to be concentrated.

(a) two particles: Consider a system of two particles of masses $m_1$ and $m_2$ located at position vectors $\vec{r}_1$ and $\vec{r}_2$ respectively, with respect to the origin. The position vector of their centre of mass, $\vec{R}_{CM}$, is given by:

$$\vec{R}_{CM} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1 + m_2}$$

If the coordinates of the particles are $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the coordinates of the centre of mass are:

$$X_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}, \quad Y_{CM} = \frac{m_1y_1 + m_2y_2}{m_1 + m_2}, \quad Z_{CM} = \frac{m_1z_1 + m_2z_2}{m_1 + m_2}$$

(b)$n$-particle system: Consider a system of $n$ particles with masses $m_1, m_2, \dots, m_n$ located at position vectors $\vec{r}_1, \vec{r}_2, \dots, \vec{r}_n$. The position vector of the centre of mass, $\vec{R}_{CM}$, is given by:

$$\vec{R}_{CM} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \dots + m_n\vec{r}_n}{m_1 + m_2 + \dots + m_n}$$

This can be written in summation notation as:

$$\vec{R}_{CM} = \frac{\sum_{i=1}^{n} m_i \vec{r}_i}{\sum_{i=1}^{n} m_i} = \frac{1}{M} \sum_{i=1}^{n} m_i \vec{r}_i$$

(where $M = \sum_{i=1}^{n} m_i$ is the total mass of the system).

Physical Significance of Centre of Mass

It simplifies the analysis of the mechanics of complex systems. The entire system can be treated as a single point particle of mass $M$ located at the centre of mass.

The motion of the centre of mass is completely independent of the internal forces acting between the particles. It only depends on the external forces applied to the system ($\vec{F}_{ext} = M\vec{A}_{CM}$).

In the absence of external forces, the centre of mass either remains at rest or moves with a constant velocity.

Q2. Show that the velocity of Centre of mass of a moving isolated system remains constant in the absence of external force.

Answer

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 $F_{ext}=\:f_{1}+f_{2}+f_{3}+\cdots+f_{n}$ $$F_{ext}=\:Ma_{cm}$$ For an isolated system, the net external force is zero ($\vec{F}_{ext} = 0$). $$0 = M \frac{d\vec{V}_{cm}}{dt}$$ Since mass $M$ cannot be zero: $$\frac{d\vec{V}_{cm}}{dt} = 0$$ $$\vec{V}_{cm} = \text{constant}$$ This shows that in the absence of an external force, the velocity of the centre of mass of an isolated system remains constant.

Q3. There is no effect on the state of centre of mass in the absence of external forces. Explain it with an example.

Answer

Ans. The net external force $\vec{F}_{ext} = M\vec{A}_{CM}$.

If the net external force is zero ($\vec{F}_{ext} = 0$), the acceleration of the centre of mass is zero ($\vec{a}_{cm} = 0$). This means the velocity of the centre of mass ($\vec{V}_{cm}$) remains constant. Internal forces acting between particles within the system can change the velocities of individual particles, but they cannot alter the overall state of rest or uniform motion of the centre of mass.

Example: A spinning dumbbell on a frictionless table

When a spinning dumbbell slides across a frictionless table ($\vec{F}_{ext} = 0$), the tension connecting the masses acts solely as an internal force. Therefore, while the individual masses trace complex spirals, the dumbbell's centre of mass moves in a perfectly straight line at a constant speed.

Q4. Give four examples of Centre of mass motion.

Answer
Ans.

The Earth-Moon system

A springboard diver

A falling cat twisting in mid-air

A rolling vehicle wheel

A thrown rugby ball

Q5. If net force on a system of particles is zero, what is the value of (a) Acceleration of Centre of mass (b) Momentum of Centre of mass (c) velocity of Centre of mass and (d) velocity of an individual particles of the system.

Q6. Distinguish between Centre of mass and Centre of gravity.

Answer
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.

Q7. Show that a system of particles moving under the influence of an external force, moves as if the force is applied at its Centre of mass.

Answer

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.

Q8. Explain about the Centre of mass of earth-moon system and its rotation around the sun.

Answer

Ans. The mass of the earth is greater 81 times that of the 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.

Q9. Define angular velocity $\omega$. Derive $v=r\omega$.

Answer

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 $t$. $$\begin{aligned}
&{\frac{dx}{dt}}={\frac{d(r\theta)}{dt}} \\
&V=r{\frac{d(\theta)}{dt}} \\
&V=r\omega \\
\end{aligned}$$

Q10. Define angular acceleration and torque. Establish the relation between angular acceleration and torque.

Answer

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.
$$\begin{aligned}&\tau=r\times F\\&\tau=r\times ma \:\:\:(\because F=m\alpha)\\&\tau=r\times mr\alpha\:\:\:(\because a=r\alpha)\\&\tau=mr^{2}\times\alpha\: \\& \therefore \: \tau=I\alpha\\\end{aligned}$$

Q11. Write the angular equations of motion for a particle rotating about a fixed axis.

Answer
Ans.
The angular equations of motion for a particle rotating about a fixed axis with a constant angular acceleration ($\alpha$) are:

$\omega = \omega_0 + \alpha t$

$\theta = \omega_0 t + \frac{1}{2} \alpha t^2$

$\omega^2 = \omega_0^2 + 2\alpha\theta$

Where:

$\theta$ = Angular displacement

$\omega_0$ = Initial angular velocity

$\omega$ = Final angular velocity

$\alpha$ = Constant angular acceleration

$t$ = Time taken

Q12. Write the expressions for (a) Torque (b) Angular momentum (c) Rotational power and (d) Rotational kinetic energy.

Answer
Ans.
(a) Torque ($\tau$)

Vector form: $\vec{\tau} = \vec{r} \times \vec{F}$

Magnitude: $\tau = r F \sin\theta$(where $\theta$ is the angle between the position vector $r$ and force $F$)

(b) Angular momentum ($L$)

Vector form: $\vec{L} = \vec{r} \times \vec{p}$

Magnitude : $L = r p \sin\theta$ (or $L = m v r \sin\theta$) (where $\theta$ is the angle between the position vector $r$ and linear momentum $p$)

(c) Rotational power ($P$)

$P = \tau \omega$

(d) Rotational kinetic energy ($K_R$)

$K_R = \frac{1}{2} I \omega^2$

Q13. Write the expression for Moment of Inertia of (a) Solid sphere of radius R about an axis passing through the centre and perpendicular to the plane (b) Circular ring of radius R through its Centre and perpendicular to its plane (c) Disc of radius R through its Centre and perpendicular to its plane (d) Solid cylinder of length l and radius R through its Centre and perpendicular to its length.

Q14. Draw the Graph between (a) Moment of Inertia and radius of gyration. (b) Rotational K.E and angular velocity.

Q15. State Parallel axis theorem and give expression.

Q16. State Perpendicular axis theorem and give expression.

Long Answer Questions

Q1. State and prove the principle of conservation of angular momentum. Explain the principle of conservation of angular momentum with examples.

Q2. Derive an expression for the moment of inertia of a thin rectangular lamina about the axis of rotation is perpendicular to the plane of the lamina and passing through one of its corners.

Q3. A $50\,\mathrm{kg}$ mass is being lifted by using a $1$-metre-long lever. If the length of the lever is reduced by $20\%$ how much more or less force is to be applied to achieve the same torque.

Q4. A mechanic applies a force of $100\,\mathrm{N}$ at the end of a spanner to tighten a bolt. The length of the spanner is $0.5$ metre. How much torque is exerted on the bolt? If the mechanic increases the force applied by $20\%$ but holds the spanner closer at $0.4$ metre. Would it result in an increase or decrease of torque?

Q5. Deduce an expression for the moment of inertia of a solid sphere about a tangent using a parallel axis theorem.

Q6. Find the moment of inertia of a circular ring about a tangent (i) in its plane (ii) in a perpendicular plane.

Problems

Q1. Two spheres of masses $6\,\mathrm{kg}$ and $3\,\mathrm{kg}$ are moving with velocities $3\,\mathrm{m\,s^{-1}}$ and $6\,\mathrm{m\,s^{-1}}$ away from each other along the same line. Find the velocity of the Centre of mass.

Q2. Two $2\,\mathrm{kg}$ masses have velocities $\mathbf{v}_1=4\mathbf{i}+2\mathbf{j}$ and $\mathbf{v}_2=2\mathbf{i}+3\mathbf{j}$. Find the velocity of Centre of mass.

Q3. A flywheel of moment of inertia $50\,\mathrm{kg\,m^2}$ and radius $2\,\mathrm{m}$ is accelerated by applying a tangential force of magnitude $10\,\mathrm{N}$. Find the angular velocity after the $4$ seconds from the start.

Q4. An acrobat spins about a vertical axis at $160\,\mathrm{rpm}$ with his arms folded. By folding hands further, the moment of inertia about the axis of rotation decreases by $20\%$. Calculate the new rate of rotation.

Q5. Three particles each of mass $100\,\mathrm{g}$ are placed at the vertices of an equilateral triangle of side length $10\,\mathrm{cm}$. Find the moment of inertia of the system about an axis passing through the centroid of the triangle and perpendicular to its plane.

Q6. Four particles each of mass $100\,\mathrm{g}$ are placed at the vertices of a square of side $10\,\mathrm{cm}$. Find the moment of inertia of the system about an axis passing through the Centre of the square and perpendicular to its plane. Find also the radius of gyration of the system.

Q7. Determine the kinetic energy of a circular disc rotating with a speed of $60\,\mathrm{rpm}$ about an axis passing through a point on its circumference and perpendicular to its plane. The circular disc has a mass of $5\,\mathrm{kg}$ and radius $1\,\mathrm{m}$.

Q8. The moment of inertia of a flywheel making $300$ revolutions per minute is $0.3\,\mathrm{kg\,m^2}$. Find the torque required to bring it to rest in $20$ sec.

Q9. When $100\,\mathrm{J}$ of work is done on a flywheel, its angular velocity is increased from $60\,\mathrm{rpm}$ to $180\,\mathrm{rpm}$. What is the moment of inertia of the wheel.

Q10. A wheel is rotating with a speed of $500\,\mathrm{rpm}$ on a shaft. Second identical wheel, initially at rest is suddenly coupled on the same shaft. What is the speed of the resultant combination?

Q11. The radius of gyration of a uniform thin rod of length $L$ about an axis perpendicular to the rod and passing through a point on the rod is equal to the radius of gyration of the rod about its centre. Find the distance about this axis from centre of the rod.

Q1. Find the centre of mass of three particles at the vertices of an equilateral triangle. The particles have masses $100\,\mathrm{g}$, $150\,\mathrm{g}$ and $200\,\mathrm{g}$, and the side of the equilateral triangle is $0.5\,\mathrm{m}$.

Q2. Find the centre of mass of a triangular lamina.

Q3. Find the centre of mass of a uniform L-shaped lamina (a thin flat plate) shown in Fig. 6.12. The mass of the lamina is $3\,\mathrm{kg}$.

Q4. A $3\,\mathrm{m}$ long ladder weighing $20\,\mathrm{kg}$ leans on a frictionless wall. Its feet rest on the floor $1\,\mathrm{m}$ from the wall as shown in Fig. 6.15. Find the reactions of the wall and the floor.

Q5. Consider the radioactive decay of a moving unstable particle, like the muon, which is a lepton. A muon disintegrates into an electron and two neutrinos. Consider the conservation of linear momentum and the motion of the centre of mass of the system.

Q6. Obtain Eq. (6.24) from the corresponding equation for translational motion.

Q7. The angular speed of a motor wheel is increased from $1200\,\mathrm{rpm}$ to $3120\,\mathrm{rpm}$ in $16\,\mathrm{s}$. What is the angular acceleration assuming it to be uniform? How many revolutions does the wheel make during this time?

Q8. Find the torque of a force $7\mathbf{i}+9\mathbf{j}-5\mathbf{k}$ about the origin. The force acts on a particle whose position vector is $\mathbf{i}-\mathbf{j}+\mathbf{k}$.

Q9. Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.

Q10. Two forces $F_1$ and $F_2$ constitute a couple. Show that the moment of a couple is independent of the point about which moments are taken.

Q11. What is the moment of inertia of a disc about any one of its diameters?

Q12. What is the moment of inertia of a rod of mass $M$, length $l$ about an axis perpendicular to it through one end?

Q13. What is the moment of inertia of a ring about a tangent to the ring in its plane?

Q14. A cord of negligible mass is wound round the rim of a flywheel of mass $20\,\mathrm{kg}$ and radius $20\,\mathrm{cm}$. A steady pull of $25\,\mathrm{N}$ is applied to the cord. Find (a) the angular acceleration of the wheel, (b) the work done by the pull when $2\,\mathrm{m}$ of the cord is unwound, (c) the final kinetic energy of the wheel at this point, assuming that the wheel starts from rest, and (d) compare answers to parts (b) and (c).