Class 11th

A hollow sphere (mass m, radius R) is put inside a hollow cone (mass m, radius 3R and semi-angle $30°$ ) as shown in the figure. The whole arrangement is rigidly fixed to the ground at the apex of the cone. Both the sphere and the cone are made of the same material (a = 10^-4/ $°C$ ). Considering that major changes in dimensions could happen only due to heating effects, find the increase in potential energy (in Joule) of the system if it's temperature is increased by   $100°C.$

As a train moving with constant speed crosses the platform, the apparent frequency heard by a person standing on the platform during approach and recession differ by 2% of the actual frequency of the horn. If the speed of sound is 330 m/s, find the approximate speed of the train.

A cylindrical metal rod BC of length 1 m and radius 1 cm is joint in series to another rod AB of same material and length but having twice the radius as shown in the figure.

End A is placed at a temperature of 100°C and end C is placed at 50°C temperature. Find the temperature of the junction B.

Two particles start moving from the same point in two different directions. The first particle moves along the positive x-axis with a constant acceleration of 3 m/s². The second particle moves in a straight line making an angle of 60° with the positive x-axis with a constant speed of 24 m/s. Find the time after which the relative velocity of the particles is minimum.

A vernier calliper whose 10 vernier scale divisions coincide with 9 main scale divisions is used to take a reading. Each main scale division is 1 mm. Before taking the reading it is observed that the 4^th division of the vernier scale coincides with the main scale and zero of main scale is behind zero of vernier scale. When the reading is taken using this vernier calliper, the main scale reads 2.5 cm and the 7^th division of the vernier scale coincides with the main scale. Find the final value of the measurement.

A massless metal plate is placed on a horizontal tabletop lubricated with oil. The sheet is a square of side length $\mathcal{l}$ = 1.0m and the oil layer has thickness h = 1.0mm. Initially one edge of the sheet coincides with one edge of the table. The sheet is pulled outwards without rotation with a constant force F = 15 N. If coefficient of viscosity of the oil is $\eta$ = 0.2 N-s/m², how long (in second) will it take to pull half of the sheet out of the table?

Two particles are executing SHM which are described by the equations x₁ = asin(ωt + π/6) and x₂ = asin(ωt + π/4). Find the minimum time interval between the particles to cross their mean position.

A particle of mass 2 kg is moving in the direction of positive x-axis. Initial velocity of the particle is 1 m/s. The instantaneous power vs time graph of the particle is shown in the figure. Find the velocity of the particle at 7 seconds.

A string of mass per unit length $\mu$  is clamped at both ends such that one end of the string is at x = 0 and the other end at x = L. When string vibrates in fundamental mode, amplitude of the midpoint of string is a and tension is string is F. Find the total oscillation energy (in J) stored in the string.

(Use L = 1m, F = 10 N, a =   $\frac{1}{\pi }m)$

In an imaginary electric field, the electric potential energy of an electron at a distance r from the centre of force field is given by U = kr², where k is a positive constant of appropriate dimensions. If the electron is moving in a circular orbit of radius r about the centre, then the orbital time period of the electron is proportional to: