Tag: stability and centre of mass

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Questions Related to stability and centre of mass

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

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

  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. $\left( 2\hat { i } +2\hat { j } +2\hat { k } \right) m{ s }^{ -1 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $\vec {v}=\dfrac {20\hat i-70\hat k+30\hat j+(-20\hat i+70\hat k-6\hat j)}{10+2}$
$\Rightarrow \ \boxed {\vec {v}=\dfrac {24\ \hat j}{12}=2\hat j\ m/s}$
Multiple choice physics turning effects of forces stability and centre of mass center of mass centre of mass

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  

  1. $12$

  2. $16$

  3. $20$

  4. $22$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a cubical box of side L = 40 cm open at the top, the mass is distributed over 5 faces (bottom + 4 sides). The center of mass of the bottom is at z=0, and the center of mass of the 4 sides is at z=L/2 = 20 cm. Using the weighted average: z_cm = (1*0 + 4*20) / 5 = 80 / 5 = 16 cm.

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

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

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

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

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

  4. $\dfrac { 4R }{ 3\pi } $
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

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 

  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
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The moment of inertia depends on the distribution of mass relative to the axis. With only three particles at (a,0,0), (0,a,0), and (0,0,a), the mass distribution is not symmetric enough to determine the Z-axis moment of inertia solely from the X and Y values without knowing the masses or the full body configuration.

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

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

  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.
Reveal answer Fill a bubble to check yourself
E Correct answer
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

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

The centre of a mass of a rigid body lies

  1. inside the body

  2. outside the body

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

  4. either $(a)$ or $(b)$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

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 :

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $\dfrac{R}{2}=\dfrac{m}{8}$

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