A hoop of radius r and mass m rotating with an angular velocity ω_{0}
is placed on a rough horizontal surface.The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?
rω_{0}/4
rω_{0}/3
rω_{0}/2
rω_{0}
C.
rω_{0}/2
From conservation of angular momentum about any fixed point on the surface
A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc
remains unchanged
continuously decreases
continuously increases
first increases and then decreases
D.
first increases and then decreases
A metallic rod of length ‘l’ is tied to a string of length 2l and made to rotate with angular speed ω on a horizontal table with one end of the string fixed. If there is a vertical magnetic field ‘B’ in the region, the e.m.f. induced across the ends of the rod is
D.
Two identical charged spheres suspended from a common point by two massless strings of length l are initially a distance d(d < < l) apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate.As a result the charges approach each other with a velocity v. Then as a function of distance x between them
v ∝ x^{-1}
v ∝ x^{1/2}
v ∝ x
v ∝ x^{-1/2}
D.
v ∝ x^{-1/2}
A diatomic molecule is made of two masses m_{1} and m_{2} which are separated by a distance r. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by (n is an integer)
D.
Rotational kinetic energy of the two body system rotating about their centre of mass is
A cylindrical tube, open at both ends, has a fundamental frequency, f, in the air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now
m_{1}r_{1}:m_{2}r_{2}
m_{1} :m_{2}
r_{1} :r_{2}
1:1
C.
r_{1} :r_{2}
As their period of revolution is same, so its angular speed is also same. Centripetal acceleration is circular path,
a= ω^{2}r
Thus,
A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed ω rad/s about the vertical. About the point of suspension
angular momentum is conserved
angular momentum changes in magnitude but not in the direction
angular momentum changes in direction but not in magnitude
angular momentum changes both in direction and magnitude
C.
angular momentum changes in direction but not in magnitude
This question has Statement I and Statement II. Of the four choices given after the Statements, choose the
one that best describes the two Statements.
Statement – I: A point particle of mass m moving with speed v collides with stationary point particle of mass M. If the maximum energy loss possible is given as f
Statement – II : Maximum energy loss occurs when the particles get stuck together as a result of the collision.
Statement – I is true, Statement – II is true, Statement – II is a correct explanation of Statement – I.
Statement – I is true, Statement – II is true, Statement – II is not a correct explanation of Statement – I.
Statement – I is true, Statement – II is false.
Statement – I is false, Statement – II is true
D.
Statement – I is false, Statement – II is true
Before collision, the mass is m and after collision, the mass is m+M
therefore, Maximum energy loss
=
From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is :
h^{2}/4R
3h/4
5h/8
3h^{2}/8R
B.
3h/4
We know that centre of mass of uniform solid cone of height h is at height h/4 from base therefore,
h-z_{o} = h/4
z_{o} = h-h/4 = 3h/4
From a solid sphere of mass M and radius R, a cube of the maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is
C.
Consider the cross-sectional view of a diametric plane as given the figure.
Using geometry of the cube
Volume density of the solid sphere
Moment of inertia of the cube about the given axis is