A spherical cavity of radius r is curved out of a uniform solid s… - Vidyadip

MCQ

A spherical cavity of radius $r$ is curved out of a uniform solid sphere of radius $R$ as shown in the figure below. The distance of the centre of mass of the resulting body from that of the solid sphere is given by

A
$\frac{R-r}{2}$
B
$\frac{R+r}{2}$
C
$0$
✓
$\frac{r^3}{R^2+R r+r^2}$

✓

Answer

Correct option: D.

$\frac{r^3}{R^2+R r+r^2}$

d
(d)

Centre of mass of solid sphere is at origin (centre of sphere).

Now, centre of mass of sphere with cavity is calculated as follows.

$x_{ CM }=\frac{m_1 x_1-m_2 x_2}{m_1-m_2}$

$=\frac{\left(\frac{4}{3} \pi R^3 \cdot \rho\right)(0)-\left(\frac{4}{3} \pi r^3 \cdot \rho\right)(R-r)}{\frac{4}{3} \pi R^3 \rho-\frac{4}{3} \pi r^3 \rho}$

where, $\rho=$ density of material of sphere.

$\Rightarrow \quad x_{ CM }=-\frac{r^3(R-r)}{\left(R^3-r^3\right)}$

$=\frac{-r^3(R-r)}{(R-r)\left(R^2+R r+r^2\right)}$

$=\frac{-r^3}{R^2+R r+r^2}$

So, distance between two mass centres is

$d=\frac{r^3}{R^2+R r+r^2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 6. system of particles and rotational motion→6. system of particles and rotational motion — all question types→Physics STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The deceleration experienced by a moving motor boat, after its engine is cut-off is given by $\frac{{dv}}{{dt}} = - k{v^3}$ , where $k$ is constant. If $v_0$ is the magnitude of the velocity at cut-off, the magnitude of the velocity at a time $t$ after the cut-off is

Identify the correct statements from the following:

$(A)$ Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.

$(B)$ Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.

$(C)$ Work done by friction on a body sliding down an inclined plane is positive.

$(D)$ Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero.

$(E)$ Work done by the air resistance on an oscillating pendulum in negative.

Choose the correct answer from the options given below:

If $\overrightarrow A = 2\hat i + 4\hat j - 5\hat k$ the direction of cosines of the vector $\overrightarrow A $ are

$Assertion$ : The driver in a vehicle moving with a constant speed on a straight road is an inertial frame of reference.

$Reason$ : A reference frame in which Newton's laws of motion are applicable is non-inertial

The equation of a wave disturbance is given as : $y = 0.02 cos \left( {\frac{\pi }{2} + 50\pi t} \right) cos (10 x),$ where $x$ and $y$ are in meters and $t$ in seconds. Choose the wrong statement:

A kettle with $2\, littre$ water at $27\,^oC$ is heated by operating coil heater of power $1\, kW$. The heat is lost to the atmosphere at constant rate $160\, J/sec$, when its lid is open. In how much time will water heated to $77\,^oC$. (specific heat of water $= 4.2\, kJ/kg$) with the lid open ?

Which one of the following statements is correct

Work done by restoring force in a string within elastic limit is $-10 \,J$. Maximum amount of heat produced in the string is .......... $J$

A projectile crosses two walls of equal height $H$ symmetrically as shown The angle of projection of the projectile is

A block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined an at an angle $\theta $ to the vertical. The minimum value of coefficient of friction between the block and the surface for which the block will remain in equilibrium is