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Stability and centre of mass - class-X

Description: stability and centre of mass
Number of Questions: 44
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Tags: physics turning effects of forces rigid body dynamics systems of particles and rotational motion centre of mass motion of system of particles and rigid bodies
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The reduce mass of two particles having masses m and 2 m is 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. 2 m

  2. 3 m

  3. $\dfrac {2 m}{3}$

  4. $\dfrac { m}{2}$

Show answer
Correct Option: C
Explanation:

Given,

$m _1=m$
$m _2=2m$

The reduced mass of two particle system is given by
$\mu=\dfrac{m _1m _2}{m _1+m _2}$

$\mu=\dfrac{m\times 2m}{m+2m}$

$\mu=\dfrac{2m}{3}$
The correct option is C.

Two bodies of masses 10 kg and 2 kg are moving with velocities $2\hat { i } -7\hat { k } +3\hat { j }\ m{ s }^{ -1 }$ and $-10\hat { i } +35\hat { k } -3\hat { j }\ m{ s }^{ -1 }$ respectively. The velocity of their centre of mass is

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $2\hat { i }\ m{ s }^{ -1 }$

  2. $2\hat { j }\ m{ s }^{ -1 }$

  3. $\left( 2\hat { j } +2\hat { k } \right) m{ s }^{ -1 }$

  4. $\left( 2\hat { i } +2\hat { j } +2\hat { k } \right) m{ s }^{ -1 }$

Show answer
Correct Option: B
Explanation:
$m _1 = 10\ kg\quad \vec {v _1}=2\hat i -7\hat k+3\hat j\ m/s$
$m _2=2\ kg \quad \vec {v _2}=-10\hat i +35\hat k-3\hat j\ m/s$
velocity of centre of mass should be
$\vec {v}=\dfrac {m _1 \vec {v _1}+m _2 \vec {v _2}}{m _1 +m _2}$
or $\vec {v}=\dfrac {20\hat i-70\hat k+30\hat j+(-20\hat i+70\hat k-6\hat j)}{10+2}$
$\Rightarrow \ \boxed {\vec {v}=\dfrac {24\ \hat j}{12}=2\hat j\ m/s}$

Figure shows a cubical box that has been constructed from uniform metal plat of negligible thickness. The box is open at the top and has edge length $40 /cm$. The $z$ co-ordinate of the centre of mass of the box in $cm$, is  

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $12$

  2. $16$

  3. $20$

  4. $22$

Show answer
Correct Option: B

The centre of mass of a uniform thin hemispherical shell of radius R is located at a distance ?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\dfrac { \pi R }{ 2 } $

  2. $\dfrac { 2R }{ 3 } $

  3. $\dfrac { R }{ 2 } $

  4. $\dfrac { 4R }{ 3\pi } $

Show answer
Correct Option: C
Explanation:

The centre of mass of a uniform thin hemispherical shell of radius R is located at a distance  $\dfrac{R}{2}$ from the center.

A body having its centre of mass at the origin has three of its particles at $(a,0,0),(0,a,0),(0,0,a)$ the moment of inertia of the body about X and Y axis are $0.2kg{m _2}$ the moment of inertia about its Z axis is 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. is $0.20kg{m _2}$

  2. is $0.40kg{m _2}$

  3. $0.20\sqrt 2 kg{m^2}$

  4. cannot be deducted with this information

Show answer
Correct Option: D

The centre of mass of a system of particles is at the origin. It follows that:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. the number of particles to the right of the origin is equal to the left of origin.

  2. the total mass of the particles to the right of the origin is same as total mass to the left of the origin.

  3. the number of particles on the X-axis should be equal to the number of particles on the Y-axis .

  4. if there is a particle on the +ve X-axis, there should be atleast one particle on the -ve X-axis.

  5. None of these.

Show answer
Correct Option: E
Explanation:

Center of mass of a system of particles is at the origin. It follows that:

(a) Number of particles to the right of origin=Number of particles to the left of origin
(b) Total mass of particles to the right of origin=Total mass of particles to the left of origin

The centre of a mass of a rigid body lies

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. inside the body

  2. outside the body

  3. neither $(a)$ nor $(b)$

  4. either $(a)$ or $(b)$

Show answer
Correct Option: D
Explanation:

Centre of mass of rigid bodies may lie either inside or outside the body.

For example:  COM of solid sphere lies inside it whereas COM of uniform circular ring lies at its geometric centre where there is actually no matter.

The point through which the total weight appears to act for any orientation of the object is ______.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. centre of gravity.

  2. centre of momentum

  3. centre of force

  4. none of the above

Show answer
Correct Option: A
Explanation:

the point where the total weight appears to act for any orientation of the object is centre of gravity .

The centre of gravity depends on the acceleration due to gravity at the given place.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

centre of gravity depends on shape of object and its mass distribution.

so the answer is B.

A uniform metal disc of radius R is taken and out of it a disc of diameter $\dfrac{R}{2}$ is cut off from the end.The centre of mass of the remaining part will be :

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\dfrac{R}{28}$ from the centre

  2. $\dfrac{R}{3}$ from the centre

  3. $\dfrac{R}{5}$ form the centre

  4. $\dfrac{R}{6}$ from the centre

Show answer
Correct Option: A
Explanation:
No correct option.

Given,

$Radius =R,D=\dfrac{R}{2}$

Let origin be the center  

$0=\dfrac{m _1x _1+m _2x _2}{m _1+m _2}$

Mass of the diameter $\dfrac{R}{2}=\dfrac{m}{8}$

$0=\dfrac{\dfrac{7m}{8}X _1+\dfrac{m}{8}(\dfrac{R}{4})}{\dfrac{7m}{8}+\dfrac{m}{8}}=\dfrac{7m}{8}x _1+\dfrac{m}{8}\times\dfrac{R}{4}\Rightarrow x _1=\dfrac{-R}{28}$

Location of centre of mass of uniform semi-circular plate of radius R from its centre is:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\dfrac{2R}{3\pi}$

  2. $\dfrac{R}{3\pi}$

  3. $\dfrac{3R}{4\pi}$

  4. $\dfrac{4R}{3\pi}$

Show answer
Correct Option: D

Four bodies of masses 1,2,3,4 kg respectively are placed at the corners of a square of side $'a'$. Coordinates of centre of mass are (take $1\ kg$ at origin, $2\ kg$ on X-axis and $4\ kg$ on Y-axis)

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\Big \lgroup \dfrac{7a}{10}, \dfrac{a}{2} \Big \rgroup$

  2. $\Big \lgroup \dfrac{a}{2}, \dfrac{7a}{10} \Big \rgroup$

  3. $\Big \lgroup \dfrac{a}{2}, \dfrac{3a}{10} \Big \rgroup$

  4. $\Big \lgroup \dfrac{7a}{10}, \dfrac{3a}{2} \Big \rgroup$

Show answer
Correct Option: C

Two blocks of masses $8$kg are connected by a spring of negligible mass and placed on a frictions less horizontal surface. An impulse gives a velocity of $12$m/s to the heavier block in the direction of lighter block. The velocity of the center of mass is:-

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $12$m/s

  2. $10$m/s

  3. $8$m/s

  4. $6$m/s

Show answer
Correct Option: D
Explanation:

Velocity of center of mass is $=\dfrac{m _1v _1+m _2v _2}{m _1+m _2}$


                                                $=\dfrac{8\times 12+8\times 0}{8+8}$


                                                $=\dfrac{96+0}{16}$

                                                $=6m/s$
Hence, the answer is $6m/s.$

A solid cylinder at rest at the top of an inclined plane of height 2.7 m rolls down without slipping. If the same cylinder has to slide down a frictionless inclined plane and acquire the same velocity as that acquired by the centre of mass of the rolling cylinder at the bottom of the inclined plane, the height of the inclined plane in meters should be 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. 2.2

  2. 1.2

  3. 1.6

  4. 1.8

Show answer
Correct Option: C

Find the coordination of center of mass of a uniform semicircle closed wire frame with respect to the origin which is at its center.The radius of the circular portion is R.                

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\left( {\dfrac{{4R}}{{3\pi }},0} \right)$

  2. $\left( {\dfrac{{2R}}{{\pi }},0} \right)$

  3. $\left( {\dfrac{R}{{\pi + 2}},0} \right)$

  4. $\left( {\dfrac{2R}{{\pi + 2}},0} \right)$

Show answer
Correct Option: A
Explanation:

We know that, $x+cm=\cfrac{4}{\pi R^2}\int^0 _R{-t^2 dt}$

$=\cfrac{4}{\pi R^2}|-\cfrac{t^3}{R}|^0 _R=\cfrac{4}{\pi R^2}(\cfrac{R^3}{3})=\cfrac{4R}{3\pi}$
Thus, co-ordinates should be $=[\cfrac{4}{3\pi},0]$

Two particles having mass ratio n : 1 are interconnected by a light in extensible string that passes over a smooth pulley. If the system is released, then the acceleration of the centre of mass of the system is

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $( n - 1 ) ^ { 2 } g$

  2. $\left( \frac { n + 1 } { n - 1 } \right) ^ { 2 } g$

  3. $\left( \frac { n - 1 } { n + 1 } \right) ^ { 2 } g$

  4. $\left( \frac { n + 1 } { n - 1 } \right) 9$

Show answer
Correct Option: C
Explanation:

Given$,$

$\frac{{{m _1}}}{{{m _2}}} = \frac{n}{1} = n$
Each mass have the acceleration $a = \frac{{\left( {{m _1} - {m _2}} \right)}}{{{m _1} + {m _2}}}$
however ${{m _1}}$ which is heavier will have the will have acceleration ${{a _1}}$ vertically down while the lighter mass ${{m _2}}$ will have acceleration ${{a _2}}$ vertically up $ \to {a _2} =  - {a _1}$
The acceleration or the centre of mass of the system$,$ ${a _{cm}} = \frac{{{m _1}{a _1} + {m _2}{a _2}}}{{{m _1} + {m _2}}}$
given that ${a _2} =  - {a _1} \to {a _{cm}} = \frac{{\left( {{m _1} - {m _2}} \right){a _1}}}{{{m _1} + {m _2}}} = \frac{{{m _1} - {m _2}}}{{{m _1} + {m _2}}} \times \frac{{\left( {{m _1} - {m _2}} \right)g}}{{{m _1} + {m _2}}} = \frac{{{{\left( {{m _1} - {m _2}} \right)}^2}g}}{{{m _1} + {m _2}}}$
Since $\frac{{{m _1}}}{{{m _2}}} = n$ diving by ${{m _2}}$ and simplifying 
$ \Rightarrow {a _{cm}} = {\left( {\frac{{n - 1}}{{n + 1}}} \right)^2}g$
Hence,
option $(C)$ is correct answer.

A thin uniform rod of length l and m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\dfrac{1}{6} \dfrac{l\omega}{g}$

  2. $\dfrac{1}{2} \dfrac{l^2\omega^2}{g}$

  3. $\dfrac{1}{6} \dfrac{l^2\omega^2}{g}$

  4. $\dfrac{1}{3} \dfrac{l^2\omega^2}{g}$

Show answer
Correct Option: C
Explanation:

At lowest height

$E=\cfrac{1}{2}I \omega^2$
At maximum weight $\omega=0$
$E=mgh+0$ (his height from lower point)
By conservation of energy
$\cfrac{1}{2}I\omega^2=mgh\h=\cfrac{I\omega^2}{2mg}\h=\cfrac{ml^2\omega^2}{6mg}=\cfrac{l^2\omega^2}{6g}$

Six identical particles each of mass $m$ are arranged at the corners of a regular hexagon of side length $a$. If the mass of one of the particle is doubled, the shift in the centre of mass is

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $a$

  2. $\dfrac {6a}{7}$

  3. $\dfrac {a}{7}$

  4. $\dfrac {a}{\sqrt {3}}$

Show answer
Correct Option: C

Centre of mass is a point 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Which is geometric centre of a body

  2. From which distance of particles are same

  3. Where the whole mass of the body is supposed the

  4. none of these

Show answer
Correct Option: C

The centre of mass of a rigid body always lies inside the body. Is this statement true or false?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. True

  2. False

Show answer
Correct Option: B
Explanation:
No it's not necessary that the centre of mass of a body should lie inside the body. Consider a circular ring, its centre of mass lies at the center of the ring where there is no content of the body. So it can also lie outside the body .

A football rolls through the ground. The path followed by center of mass of football is:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. linear

  2. circular

  3. rotational

  4. all the above

Show answer
Correct Option: A
Explanation:

When a football rolls through the ground, the path followed by center of mass of ball is linear as the center of mass remains always at a fixed height from the ground when the football rolls and moves in a straight line if the ball does not changes its direction of rolling which is assumed in this case.

The correct option is A.

which of these represent the centre of mass for a semicircular ring ?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $0$

  2. $\dfrac { 4R }{ 3\pi } $

  3. $\dfrac{R}{2}$

  4. $\dfrac { 2R }{ \pi } $

Show answer
Correct Option: B

A flexible chain of length 2m and mass 1 kg initially held in vertical position such that its lower end just touches a horizontal surfaces, is released from rest at time t=0, Assuming that any part of chain which strike the plane immediately comes to rest and that the portion of chain lying on horizontal surface does not form  any heap, the height of its center of mass above surface at any instant $t=1/\sqrt { 5 } $(before it completely comes to rest) is

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. 1 m

  2. 0.5 m

  3. 1.5 m

  4. 0.25 m

Show answer
Correct Option: A

A metallic ball has spherical cavity at its centre. If the ball is heated, what happens to the cavity?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. its volume increases

  2. its volume decreases

  3. its volume remains unchanged

  4. its volume may decreased or increase depending upon the nature of metal

Show answer
Correct Option: A

A body having it's center of mass  at the origin. Then,

(The question having a multiple answers).

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. x co-ordinates of the particles may be all positive.

  2. total KE must be conserved.

  3. total KE must very.

  4. total momentum shall vary.

Show answer
Correct Option: C,D
Explanation:

If all the particles have positive x  co-ordinates then their COM can't be at the origin. Total KE may not be conserved because of internal forces. Its not given that there is no external force acting on the system, so its momentum may also change.

Where will be the centre of mass on combining two masses $m$ and $M(M>m)$ ?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $Towards \ m$

  2. $Towards \ M$

  3. $ exactly \ between \ m \ and \ M $

  4. $None \ of \ the \ above$

Show answer
Correct Option: B
Explanation:

As we can see that the center of mass will be at $r _c=\dfrac{mr+MR}{m+M}$,


where $r$ and $R$ are the position of the masses $m$ and $M$ respectively.

From the formula we notice that it will be between $m$ and $M$ and $nearer $ to the higher mass $M$.

A body has its center of mass at the origin. The x-axis coordinates of the particles :

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. may be all positive

  2. may be all negative

  3. should be all at zero

  4. may be positive for some case and negative in other cases.

Show answer
Correct Option: D
Explanation:

All co-ordinates positive will make the co-ordinate of com positive.
All co-ordinates negative will make the co-ordinates of com negative.
If all the co-ordinates are zero(0), co-ordinate of com is zero(0).
If  co-ordinate of com can be zero,  co-ordinates of some are positive and co-ordinates of some are negative.

If the linear density of a rod of length L varies as $\lambda =A+B _x$, compute its centre of mass.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $[\cfrac {L(3A+2BL)}{3(2A+BL},0,0]$

  2. $[0,\cfrac {(3A+2B)L}{(2A+3L},\cfrac L 2]$

  3. $[0,0\cfrac {L(3A+2BL)}{3(2A+BL}]$

  4. $[\cfrac L 2,00]$

Show answer
Correct Option: A

If a square of side $\dfrac{R}{2}$ is removed from a uniform circular disc of radius R as shown in the figure, the shift in centre of mass is 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\dfrac{R}{4 \pi - 1}$

  2. $\dfrac{R}{2(4 \pi - 1)}$

  3. $\dfrac{R}{3(4 \pi - 1)}$

  4. $\dfrac{R}{4(4 \pi - 1)}$

Show answer
Correct Option: B

Which of the following is not correct about centre of mass ?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. It depends on the choice of frame of reference

  2. In centre of mass frame, momentum of a system is always zero

  3. Internal forces may affect the motion of centre of mass

  4. Centre of mass and centre of gravity coincide in uniform gravitational field

Show answer
Correct Option: C
Explanation:

The internal forces all balance each other so that there is no change in the mass distribution of the body, thus center of mass remains unchanged. Rest all options are correct.

A point at which a whole weight of body act vertically downward is ________.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. centre of gravity

  2. centre of mass

  3. centre of force

  4. centre of acceleration

Show answer
Correct Option: A

In classical system:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. the varying mass system is not considered

  2. the varying mass system must be considered

  3. the varying mass system may be considered

  4. only varying of mass due to velocity is considered

Show answer
Correct Option: C

Statement 1: When we lean behind over the hind legs of the chair, the chair falls back after a certain angle.

Statement 2: Centre of mass lying outside the system makes the system unstable.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Statement 1 is false, Statement 2 is true

  2. Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

  3. Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1

  4. Statement 1 is true, Statement 2 is false

Show answer
Correct Option: D
Explanation:
The chair falls down after a certain angle because it becomes unstable as the normal reaction by ground on chair becomes zero.
There are several mechanical structure whose COM lies outside its body but they are quite stable.

So option D is correct.

On a stationary sail boat, air is blown at the sails from a fan attached to the boat. The boat will

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. remain at rest

  2. spin round

  3. move in the direction in which air is blown

  4. move in a direction opposite to that in which air is blown

Show answer
Correct Option: A
Explanation:

the boat will remain at the rest  because the thrust or recoil force of fan at boat and force of air on the sails will cancel each other and boat will not at all move .

so the answer is A.

The passengers in a boat are not allowed to stand because: 

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. This will raise the centre of gravity and the boat will be rocked

  2. This will lower centre of gravity and the boat will rocked

  3. The effective weight of system increases

  4. Of surface tension effects

Show answer
Correct Option: A
Explanation:

when passengers stands up on the boat from the sitting situation , the centre of gravity moves upwards and it increases the chances of toppling of boat .

so the answer is A.

State whether true or false.
The centre of gravity of regular shaped objects is at their geometric centre.
physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

The given statement is true that centre of gravity of rectangular shaped objects is at their geometric centre.

The point through which the total weight of an object appears to act for any orientation of the object is ______.

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Centre of buoyancy

  2. Centre of gravity

  3. Centre of curvature

  4. Median of a triangle

Show answer
Correct Option: B
Explanation:

The point through which the total weight of an object appears to act for any orientation of the object is centre of gravity.

State whether true or false.
The position of centre of gravity of an object depends on the acceleration due to gravity at the given place.
physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. True

  2. False

Show answer
Correct Option: B
Explanation:
The center of gravity is a geometric property of any object. The center of gravity is the average location of the weight of an object with mass.
Therefore the position of centre of mass has nothing to do with the acceleration due to gravity.
So, the statement is FALSE

The centre of mass of a body:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Lies always at the geometrical center

  2. Lies always inside the body

  3. Lies always outside the body

  4. Lies within or outside the body

Show answer
Correct Option: D
Explanation:

The centre of mass of a body can lie within or outside the body.

For example, centre of mass of a uniform rod lies at its geometrical centre which lies within the rod whereas centre of mass of a uniform ring lies at its geometrical centre which lies outside the ring.

Two unequal masses are tied together with a cord with a compressed spring in between.
Which of the following energies is conserved for the system?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Kinetic energy

  2. Potential energy

  3. Mechanical energy

  4. None of these

Show answer
Correct Option: C
Explanation:
Both KE and PE are unserversed.
Hence, the answer is mechanical energy.


Two unequal masses are tied together with a cord with a compressed spring in between.
Which one is correct?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. Both masses will have equal KE.

  2. Lighter block will have greater KE.

  3. Heavier block will have greater KE.

  4. None of above answers is correct.

Show answer
Correct Option: B
Explanation:
Lighter block will have greater kinetic energy to lighter block will have higher velocity mass so Heavier block, hence by equation $\dfrac{1}{2}mv^2,$ the lighter will have greater KE.
Hence, the answer is Lighter block will have greater KE.

A string is wrapped around a cylinder of mass $M$ and radius $R$. The string is pulled vertically upwards to prevent the centre of mass from falling as the cylinder unwinds the string, The work done on the cylinder for reaching an angular speed $\omega$ is:

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\cfrac { 2M{ R }^{ 2 }{ \omega }^{ 2 } }{ 3 } $

  2. $\cfrac { M{ R }^{ 2 }{ \omega }^{ 2 } }{ 3 } $

  3. $\cfrac { M{ R }^{ 2 }{ \omega }^{ 2 } }{ 2 } $

  4. $\cfrac { M{ R }^{ 2 }{ \omega }^{ 2 } }{ 4 } $

Show answer
Correct Option: D
Explanation:
Work done is the rotational KE acquired be cylinder,
$=\dfrac{1}{2} I\omega ^2$
$=\dfrac{1}{2}\dfrac{MR^2}{2}\omega ^2$
$=\dfrac{MR^2}{4}\omega ^2.$
Hence, the answer is $\dfrac{MR^2}{4}\omega ^2.$

A straight rod of length L has one of its ends at the origin and the other at $x=L$. If the mass per unit length of the rod is given by Ax where A is constant, where is its mass centre?

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $L/3$

  2. $L/2$

  3. $2L/3$

  4. $3L/4$

Show answer
Correct Option: B
Explanation:
I assume you meant to say "a is a CONSTANT".

xc = coordinate of center of mass

M = total mass

$xc = ∫xdm / ∫dm = ∫xdm / M$

Given:$ m(x) = ax ⇒ dm/dx = a ⇒ dm = adx$

$∫xdm = ∫01 x(adx) = a/2$

$M = ∫dm = ∫(dm/dx)dx = ∫01 adx = a$

$xc = (a/2)/a = 1/2$

By the way, this is a mechanics problem (in statics), not a thermodynamics problem.

 

A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs coincide. The centre of mass of the new disc is $\alpha R$ fromthe centre of the bigger disc. The value of $\alpha$ is

physics turning effects of forces stability and centre of mass center of mass centre of mass

  1. $\cfrac{1}{2}$

  2. $\cfrac{1}{6}$

  3. $\cfrac{1}{4}$

  4. $\cfrac{1}{3}$

Show answer
Correct Option: D
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