A thin plate of mass M, length L, and width 2d is mounted vertically on a frictionless axle along the z-axis, as shown. Initially the object is at rest. It is then tapped with a hammer to provide a torque , which produces an angular impulse H about the z-axis of magnitude What is the angular speed w of the plate about the z-axis after the tap?

Three equal masses m are rigidly connected to each other by massless rods of length l forming an equilateral triangle, as shown above. The assembly is to be given an angular velocity about an axis perpendicular to the triangle. For fixed , the ratio of the kinetic energy of the assembly for an axis through B compared with that for an axis through A is equal to

A uniform rectangular solid of side length L and width w placed on a frictionless incline of variable angle, . Assume that there is a very tiny peg preventing the box from sliding down the incline at the point labeled P. At what angle of incline will the box begin to tip over?

Two particles A and B of mass 5 kg and 9 kg respectively rest on the smooth faces of a fixed wedge as shown in the figure. They are connected by a light inextensible string passing over a smooth pulley at C and are released from rest from the position shown in the diagram. In the subsequent motion B hits the ground and does not rebound. Find

A chain of length L weighing l per unit length begins to fall through a hole in the ceiling.

The velocity of the cart at time t

A rod of length 1000 mm and coefficient of linear expansion a = 10–4 per degree is placed symmetrically between fixed walls separated by 1001 mm. The Young’s modulus of the rod is 1011 N/m2 . If the temperature is increased by 20°C, then the stress developed in the rod is

If coefficient of friction between the two blocks is equal to 0.4, then magnitude of initial slope (in S.I. unit) versus t (in SI unit) will be:

If the angles of a triangle are in ratio 4 : 1 : 1 then the ratio of the longest side and perimeter of triangle is -

So, the ratio of the longest side and perimeter of triangle is

In a triangle PQR as shown in figure given that x : y : z :: 2 : 3 : 6, then the value of QPR is -

If the sides be a, b, c then (r + r₁) tan+ (r + r₂) tan+ (r + r₃) tan =

If in a triangle ABC; AD, BE and CF are the altitudes and R is the circum-radius, then the radius of the circle DEF is -

Therefore, the radius of the circle DEF is R/2.

In an equilateral triangle of side cms, the circum radius is -

The circumradius is 2cm.

If the lines and are concurrent then

is a unit vector such that 0 then a unit vector perpendicular to

The reflection of the point (2, –1, 3) in the plane 3x – 2y – z = 9 is :