A uniform flat circular rim is balanced on a sharp vertical nail by supporting it at a point A, as shown in Fig 1.41 Mark the position of the centre of gravity of the rim in the diagram by the letter G.

A non-uniform rod AB of weight w is supported horizontally in a vertical plane by two light strings PA and QB as shown in the figure.G is the centre of gravity of the rod. If PA and QB make angles 30^(@) and 60^(@) respectively with the vertical, the ratio ("AG")/("GB") is

A uniform rod of weight F_(g) and length L is supported at its ends by a frictionless through as shown in figure. (a) Show that the centre of gravity of the rod must be vertically over point O when the rod is in equilibrium. (b) Determine the equilibrium value of the angle theta .

A particle of charge q and mass m is projected with a velocity v_0 toward a circular region having uniform magnetic field B perpendicular and into the plane of paper from point P as shown in Fig. 1.136. R is the radius and O is the center of the circular region. If the line OP makes an angle theta with the direction of v_0 then the value of v_0 so that particle passes through O is

Two thin convex lenses of focal lengths f_(1) and f_(2) are separated by a horizontal distance d (where dltf_(1) , dltf_(2) ) and their centres are displaced by a vertical separation triangle as shown in the fig. Taking the origin of coordinates O, at the centre of the first lens the x and y coordinates of the focal point of this lens system, for a parallel beam of rays coming form the left, are given by:

Two thin convex lenses of focal lengths f_(1) and f_(2) are separated by a horizontal distance d (where dltf_(1) , dltf_(2) ) and their centres are displaced by a vertical separation triangle as shown in the fig. Taking the origin of coordinates O, at the centre of the first lens the x and y coordinates of the focal point of this lens system, for a parallel beam of rays coming form the left, are given by:

A uniform disc of mass M and radius R is supported vertically by a pivot at its periphery as shown. A particle of mass M is fixed to the rim and raised to the highest point above the center. The system is released from rest and it can rotate about pivot freely. The angular speed of the system when the attached object is directly beneath the pivot, is

A square card board is suspended by passing a pin through a narrow hole at its one corner. Draw a diagram to show its rest position. In the diagram, mark the point of suspension by the letter S and the centre of gravity by the letter G.

A thin horizontal circular disc is roating 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.

A wooden plank of length 1m and uniform cross-section is hinged at one end to the bottom of a tank as shown in fig. The tank is filled with water upto a hight 0.5m. The specific gravity of the plank is 0.5. Find the angle theta that the plank makes with the vertical in the equilibrium position. (Exclude the case theta=theta^@ )

A cylinder of radius R = 1 m and height H = 3 m two- thirds filled with water, is rotated about its vertical axis as shown in Fig. Determine the speed of rotation when a. the: water just starts spilling over the rim. b. the point at the centre of the base is just exposed.