An aluminium wire is wound on a piece of cork of mass m cork . The densities ρ cork , ρ al , and ρ w of cork, aluminium, and water are 0 . 5 × 10 3 kg m - 3 , 2 . 7 × 10 3 kg m - 3 , and 1 × 10 3 kg m - 3 respectively. Determine the minimum mass m al of the wire that should be wound on the cork so that the cork with the wire is completely submerge 1 in water.

The condition of complete submergence of a body is given by, M ⩾ ρ w V ; where M is the mass of the body (cork), and V is its volume. In this case, according to the question, the mass will be given by, M = m cork + m al And the volume will be, V = m cork ρ cork + m al ρ al Hence, it follows that the minimum mass of the wire is, m cork + m al = ρ w m cork ρ cork + m al ρ al m al = ρ al ρ w - ρ cork ρ al - ρ w ρ cork m cork = 2 . 7 × 10 3 1 - 0 . 5 × 10 3 2 . 7 × 10 3 - 1 × 10 3 0 . 5 × 10 3 ≃ 1 . 6 m cork .

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One rope of a swing is fixed above the other rope by $b$ . The distance between the poles of the swing is $a$ . The lengths $l_{1}$ and $l_{2}$ of the ropes are such that $l_{1}^{2} + l_{2}^{2} = a^{2} + b^{2}$ . Determine the period $T$ of small oscillations of the swing, neglecting the height of the swinging person in comparison with the above lengths.

Important Questions on Mechanics

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.1

Being a punctual man, the lift operator of a skyscraper hung an exact pendulum clock on the lift wall to know the end of the working day. The lift moves with an upward and downward accelerations during the same time (according to a stationary clock), the magnitudes of the accelerations remaining unchanged.

Will the operator finish his working day in time, or will he work more (less) than required?

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JEE Advanced

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.101

The atmospheric pressure is known to decrease with altitude. Therefore, at the upper storeys of the Moscow State University building the atmospheric pressure must be lower than at the lower storeys. In order to verify this, a student connected one arm of a

U

-shaped manometer to the upper auditorium and the other arm to the lower auditorium.

What will the manometer indicate?

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.102

Two thin-walled tubes closed at one end are inserted one into the other and completely filled with mercury. The cross-sectional areas of the tubes are

S

and

2 S

. The atmospheric pressure is

p_{0} = ρ_{mer} g h

, where

ρ_{mer}

is the density of mercury,

g

is the free-fall acceleration, and

h

is the height. The length of each tube is

l > h

.
What work

A

must be done by external forces to slowly pull out the inner tube? The pressure of mercury vapour and the forces of adhesion between the material of the tubes and mercury should be neglected.

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.103

Two cylinders with a horizontal and a vertical axis respectively rest on a horizontal surface. The cylinders are connected at the lower parts through a thin tube. The "horizontal" cylinder of radius $r$ is open at one end and has a piston in it (Fig.). The "vertical" cylinder is open at the top. The cylinders contain water which completely fills the part of the horizontal cylinder behind the piston and is at a certain level in the vertical cylinder.

Determine the level $h$ of water in the vertical cylinder at which the piston is in equilibrium, neglecting friction.

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.104

An aluminium wire is wound on a piece of cork of mass

m_{cork}

. The densities

ρ_{cork}, ρ_{al}

, and

ρ_{w}

of cork, aluminium, and

water are $0.5 \times 10^{3} kg m^{- 3}, 2.7 \times 10^{3} kg m^{- 3}$ , and $1 \times 10^{3} kg m^{- 3}$ respectively.

Determine the minimum mass $m_{al}$ of the wire that should be wound on the cork so that the cork with the wire is completely submerge $1$ in water.

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.105

One end of an iron chain is fixed to a sphere of mass $M = 10 kg$ and of diameter $D = 0.3 m$ (the volume of such a sphere is $V = 0.0141 m^{3}$ , while the other end is free. The length $l$ of the chain is $3 m$ and its mass $m$ is $9 kg$ . The sphere with the chain is in a reservoir whose depth $H = 3 m$ .

Determine the depth at which the sphere will float, assuming that iron is $7.85$ times heavier than water.

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.106

Two bodies of the same volume but of different masses are in equilibrium on a lever.

Will the equilibrium be violated if the lever is immersed in water so that the bodies are completely submerged?

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Science for Everyone Aptitude test problems in Physics>Chapter 1 - Mechanics>Exercise>Q 1.107

A flat wide and a high narrow box float in two identical vessels filled with water. The boxes do not sink when two identical heavy bodies of mass $m$ each are placed into them.

In which vessel will the level of water be higher?

Important Questions on Mechanics

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