Assuming the principal axis as the x-axis and the optical centre as the origin, how will you determine the signs of the quantities u, v and f for a lens ?

A glass wedge with a small angle of refraction theta is placed at a certain distance from a converging lens with a focal length f ,one surface of the wedge being perpendicular to the optical axis of the lens. A point sources S of light is on the other side of the lens at its focus. The rays reflected from the wedge (not from base) produce, after refraction in the lens , two images of the source displaced with respect to each other by d. Find the refractive index of the wedge glass. [Consider only paraxial rays.]

A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle theta =30degree to the axis of the lens, as shown in the figure If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x,y) at which the image is formed are

A uniform electric field pointing in positive x-direction exists in a region. Let A be the origin, B be the point on the x-axis at x=+1cm and C be the point on the y-axis at y=+1 cm. then the potetial at the points A,B and C satisfy a. V_AltV_B , b. V_AgtV_B c. V_AltV_C d. V_AgtV_C

Let y=f(x)be curve passing through (1,sqrt3) such that tangent at any point P on the curve lying in the first quadrant has positive slope and the tangent and the normal at the point P cut the x-axis at A and B respectively so that the mid-point of AB is origin. Find the differential equation of the curve and hence determine f(x).

Four point charges +8muC,-1muC,-1muC and , +8muC are fixed at the points -sqrt(27//2)m,-sqrt(3//2)m,+sqrt(3//2)m and +sqrt(27//2)m respectively on the y-axis. A particle of mass 6xx10^(-4)kg and +0.1muC moves along the x-direction. Its speed at x=+ infty is v_(0) . find the least value of v_(0) for which the particle will cross the origin. find also the kinetic energy of the particle at the origin in tyhis case. Assume that there is no force part from electrostatic force.

In an experiment to determine the focal length (f) of a concave mirror by the u-v method, a student places the object pin A on the principal axis at a distance x from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then,

OABCDE is a regular hexagon of side 2 units in the XY-plane in the first quadrant. O being the origin and OA taken along the x-axis. A point P is taken on a line parallel to the z-axis through the centre of the hexagon at a distance of 3 unit from O in the positive Z direction. Then find vector AP.

On a frictionless horizontal surface , assumed to be the x-y plane , a small trolley A is moving along a straight line parallel to the y-axis ( see figure) with a constant velocity of (sqrt(3)-1) m//s . At a particular instant , when the line OA makes an angle of 45(@) with the x - axis , a ball is thrown along the surface from the origin O . Its velocity makes an angle phi with the x -axis and it hits the trolley . (a) The motion of the ball is observed from the frame of the trolley . Calculate the angle theta made by the velocity vector of the ball with the x-axis in this frame . (b) Find the speed of the ball with respect to the surface , if phi = (4 theta )//(4) .

A circular ring of radius R with uniform positive charge density lambda per unit length is located in the y-z plane with its centre at the origin O. A particle of mass m and positive charge q is projected from the point P (Rsqrt3, 0, 0) on the positive x-axis directly towards O, with an initial speed v. Find the smallest (non-zero) value of the speed v such that the particle does not return to P.

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity omega is an example of non=inertial frame of reference. The relationship between the force vecF_(rot) experienced by a particle of mass m moving on the rotating disc and the force vecF_(in) experienced by the particle in an inertial frame of reference is vecF_(rot)=vecF_(i n)+2m(vecv_(rot)xxvec omega)+m(vec omegaxx vec r)xxvec omega . where vecv_(rot) is the velocity of the particle in the rotating frame of reference and vecr is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter fo a disc of radius R rotating counter-clockwise with a constant angular speed omega about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis (vecomega=omegahatk) . A small block of mass m is gently placed in the slot at vecr(R//2)hati at t=0 and is constrained to move only along the slot. The distance r of the block at time is