Home
Class 12
PHYSICS
A thin ring of radius R is made of a mat...

A thin ring of radius R is made of a material of density `rho` and Young's modulus Y. If the ring is rotated about its centre in its own plane with angular velocity `omega` , find the small increase in its radius.

Text Solution

Verified by Experts

Consider an element PQ of length dl. Let T be the tension and A the area of cross section of the wire.
`"Mass of element dm"="Volume"xx"density"`
`=A(dl)phi`
The component of T, towards the centre provides the necessary centripetal force
`therefore2Tsin(d theta)/(2)=(dm)Romega^(2)" ...(i)"`

For small angle, `sin(d theta)/(2)~~(d theta)/(2)~~(((dl)/(R)))/(2)`
Substitution in equation (i), we have
`2T(d theta)/(2)=(dm)R omega^(2)`
`T.(dl)/(R)=A(dl)phi R omega^(2)`
`T=Aphi omega^(2)R^(2)`
Tension in the ring is `A phi omega^(2)R^(2)`
`(DeltaR)/(R)=(T)(AY)=(phi omega^(2)R^(2))/(Y)rArr DeltaR=(phiomega^(2)R^(3))/(Y)`
Promotional Banner

Topper's Solved these Questions

  • MECHANICAL PROPERTIES OF SOLIDS

    AAKASH INSTITUTE|Exercise Try Yourself|32 Videos

  • MECHANICAL PROPERTIES OF SOLIDS

    AAKASH INSTITUTE|Exercise Assignment (SECTION - A)|31 Videos

  • MECHANICAL PROPERTIES OF FLUIDS

    AAKASH INSTITUTE|Exercise SECTION - J|9 Videos

  • Mock test 03

    AAKASH INSTITUTE|Exercise EXAMPLE|37 Videos

Similar Questions

Explore conceptually related problems

A thin ring of radius R is made of a material of density rho is Young's modulus Y . If the ring is rotated about its centre in its own plane with anglular velocity omega , if the small increase in its radius is (2rhoomega^(3)R^(3))/(lambdaY) then find lambda .

A thin ring of radius R is made of a wire of density rho and Young’s modulus Y. It is spun in its own plane, about an axis through its centre, with angular velocity w. Determine the amount (assumed small) by which its circumference increases.

A ring of radius R is made of a thin wire of material of density rho , having cross-section area a and Young's modulus y. The ring rotates about an axis perpendicular to its plane and through its centre. Angular frequency of rotation is omega . The tension in the ring will be

A ring of radius R is made of a thin wire of material of density rho , having cross-section area a and Young's modulus y. The ring rotates about an axis perpendicular to its plane and through its centre. Angular frequency of rotation is omega . The ratio of kinetic energy to potential energy is

A ring of mass m and radius R is being rotated about its axis with angular velocity omega . If a increases then tension in ring

A ring of mass m and radius R is being rotated about its axis with constant angular velocity omega in the gravity free space. Find tension in the ring.

Very thin ring of radius R is rotated about its centre. Its radius will

A ring of radius R is made of a thin wire of material of density rho having cross section area a. The ring rotates with angular velocity omega about an axis passing through its centre and perpendicular to the plane. If we consider a small element of the ring,it rotates in a circle. The required centripetal force is provided by the component of tensions on the element towards the centre. A small element of length dl of angular width d theta is shown in the figure. If T is the tension in the ring, then

A ring of radius R is made of a thin wire of material of density rho having cross section area a. The ring rotates with angular velocity omega about an axis passing through its centre and perpendicular to the plane. If we consider a small element of the ring,it rotates in a circle. The required centripetal force is provided by the component of tensions on the element towards the centre. A small element of length dl of angular width d theta is shown in the figure. If for a given mass of the ring and angular velocity, the radius R of the ring is increased to 2R , the new tension will be

A ring of radius R is made of a thin wire of material of density rho having cross section area a. The ring rotates with angular velocity omega about an axis passing through its centre and perpendicular to the plane. If we consider a small element of the ring,it rotates in a circle. The required centripetal force is provided by the component of tensions on the element towards the centre. A small element of length dl of angular width d theta is shown in the figure. The centripetal force acting on the element is