A hollow metal sphere of mass m has an instantaneous upward acceleration 'a' when released from rest fully submerged in a liquid of density $d_{1}$ . If the density of the metal is $d_{2}$ find the volume of the cavity in the sphere?

Open in App

Solution

The net force acting on the hollow metal sphere is,
$F_{n e t} = F_{b} - m g$
$m a = F_{b} - m g$
$V o l u m e (m e t a l s p h e r e) d_{2} \times a = T o t a l v o l u m e \times d_{1} \times g - v o l u m e (m e t a l s p h e r e) d_{2} g$
Write total volume in form of volume of metal+cavity.
$(V (m) + V (c)) d_{1} g = V (m) d_{2} (g + a) .$
$V (c) d_{1} g = V (m) d_{2} (g + a) - V (m) d_{1} g$
Here, $V (m) = m / d_{2}$
$V (c) d_{1} g = m (g + a) - m \frac{d_{1}}{d_{2}} g$
$V (c) = \frac{m (g + a)}{d_{1} g} - \frac{m}{d_{2}}$ .

Suggest Corrections

Similar questions

Q. A hollow metallic sphere the outer and inner diameter of which are

d_{1}

and

d_{2}

floats on the surface of the liquid. The density of the metal is

ρ_{1}

and the density of the liquid is

ρ_{2}

. What weight must be added inside the sphere in order for it to float below the level of liquid?

A body of density d and volume V floats with volume $V_{1}$ of it immersed in a liquid of density $d_{1}$ and rest of the volume $V_{2}$ immersed in another liquid of density $d_{2}$ (< $d_{1}$ ) then $V_{1}$ is

Q. A hollow metal sphere of mass

180.6 g

contains a cavity of volume

2.5 {c m}^{3}

. This metal when placed in water, displaces

24 c c

of water. Find the specific gravity of the metal.

A solid sphere of density $ρ$ and radius R is floating in a liquid of density $σ$ with half its volume submerged. When the sphere pressed down slightly and released, it executes simple harmonic motion of time period T. If viscous effect is ignored, then

Q. A solid sphere of radius R is floating in a liquid of density with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

Join BYJU'S Learning Program

A hollow metal sphere of mass m has an instantaneous upward acceleration 'a' when released from rest fully submerged in a liquid of density d1. If the density of the metal is d2 find the volume of the cavity in the sphere?

A hollow metal sphere of mass m has an instantaneous upward acceleration 'a' when released from rest fully submerged in a liquid of density $d_{1}$ . If the density of the metal is $d_{2}$ find the volume of the cavity in the sphere?