1 A particle is moving in a circular path of radius a under the action of an attractive potential $U=-\frac{k}{2r^{2}}$ , Its total energy is A) $-\frac{k}{4a^{2}}$ B) 0 C) $\frac{K}{2a^{2}}$ D) $-\frac{3k}{2a^{2}}$
2 A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the n th power of R. If the period of rotation of the particle is T, then A) $T\propto R^{\frac{3}{2}}$ For any n B) $T\propto R^{\frac{n}{2}}$ C) $T\propto R^{\frac{n}{2}+1}$ D) $T\propto R^{\frac{(n+1)}{2}}$
3 Seven identical circular planar discs, each of mass M and radius R are welded symmetrically as show in figure. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is A) $\frac{19}{2}MR^{2}$ B) $\frac{55}{2}MR^{2}$ C) $\frac{73}{2}MR^{2}$ D) $\frac{181}{2}MR^{2}$
4 The potential energy of mass m at a distance r from a fixed point O is given by $V(r)=\frac{kr^{2}}{2}$ , where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true? A) $V= \sqrt{\frac{k}{2m}}R$ B) $V= \sqrt{\frac{k}{m}}R$ C) $L= \sqrt{mk}R^{2}$ D) $L=\sqrt{\frac{mk}{2}}R^{2}$
5 A uniform capillary tube of inner radius r is dipped vertically in to a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of the water is $\sigma$ . The angle of contact between water and the wall of the capillary tube is θ . Ignore the mass of water in the meniscus. Which of the following statements is (are) true ? A) For a given material of the capillary tube, h decreases with increase in r B) For a given material of the capillary tube , h is independent of $\sigma$ C) If this experiment is performed in a lift going up with a constant acceleration, then h decreases D) h is proportional to contact angle$\theta$
6 One mole of a monoatomic ideal gas undergoes a cyclic process as shown in the figure (where ,V is the volume and T is the temperature ). Which of the statements below is (are) true? A) Process I is an isochoric process B) In process II, gas absorbs heat C) In process IV, gas releases heat D) Processes I and III are not isobaric
7 A spring block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 Nm-1 and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially ,the spring is in an unstretched condition. Another block of mass 1.0kg moving with a speed of 2.0 ms-1 collides elastically with the first block. The collision is such that the 2.0kg block does not hit the wall. The distance , in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is... A) 2.09m B) 2.5m C) 3.0m D) 1.6 m
8 In the XY-plane , the region y >0 has a uniform magnetic field $B_{1}\hat{K}$ and the region y<0 has another uniform magnetic field $B_{2}\hat{K}$ . A positively charged particle is projected from origin along the positive Y-axis with speed $v_{0}=\pi$ms-1 at t=0, as shown in figure. Neglect gravity in this problem. Let t=T be the time when the particle crosses the X-axis from below for the first time . If B2 =4B1, the average speed of the particle , in ms-1 , along the X-axis in the time interval T is........ A) 2 m/s B) 5 m/s C) 6 m/s D) 1 m/s
9 The relation between $[\epsilon_{0}]$ and $[\mu_{0}]$ A) $[\mu_{0}]= [\epsilon_{0}][L]^{2}[T]^{-2}$ B) $[\mu_{0}]= [\epsilon_{0}][L]^{-2}[T]^{2}$ C) $[\mu_{0}]= [\epsilon_{0}^{}]^{-1}[L]^{2}[T]^{-2}$ D) $[\mu_{0}]= [\epsilon_{0}^{}]^{-1}[L]^{-2}[T]^{2}$
10 A particle of mass m is initially at rest at the origin . It is subjected to a force and starts moving along the X-axis . Its kinectic energy K changes with time as $\frac{\text{d}K}{\text{d}t}=\gamma t$ , where $\gamma $ is a positive constant of appropriate dimensions. Which of the following statements is (are) true? A) The force applied on the particle is constant B) The speed of the particle is proportional to time C) The distance of the particle from the origin increases linearly with time. D) The force is conservative.
11 In an experiment to measure the speed of sound by a resonating air column, a turning fork of frequency 500Hz is used. The length of the air column is varied by changing the level of water in the resonance tube . Two successive resonances are heard at air columns of length 50.7cm and 83.9 cm . Which of the following statements is (are) true? A) The speed of sound determined from this experiment is $332 ms^{-1}$ B) The end correction in this experiment is 0.9 cm C) The wavelength of the sound wave is 66.4 cm D) The resonance at 50.7 cm corresponds to the fundamental harmonic
12 A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m= 0.4 kg is at rest on this surface . An impulse of 1.0 N s is applied to the block at time t=0, so that it starts moving along the x- axis with a velocity $v_{t}=v_{0}e^{\frac{-t}{\tau}}$ where v0 is a constant and $\tau=4$ s. The displacement of the block, in metres at $t=\tau$ is............... ( Take , e-1 =0.37) A) 6.30 B) 6.75 C) 5.60 D) 5.36
13 A ball is projected from the ground at an angle of 45° with the horizontal surface . It reaches a maximum height of 120m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy . Immediately after the bounce , the velocity of the ball makes an angle of 30° with the horizontal surface . The maximum height it reaches after the bounce, in metres, is......... A) 40 B) 25 C) 30 D) 15
14 A steel wire of diameter 0.5 mm and Young's modulus 2 × 1011 N m-2 carries a load of mass m. The length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially , the zero of vernier scale coincides with the zero of main scale . If the load on the steel wire is increased by 1.2 kg, the vernier scale division which concides with a main scale division is ........... Take , g= 10 m s-1 and π =3.2) A) 3rd Division B) 5th Division C) 6th Division D) 9th Division
15 In a photoelectric experiment , a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV . The frequency of light is just above the threshold frequency, so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100% . A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force F= n × 10-4 N due to the impact of the eletrons. The value of n is .................(Take mass of the electron, me =9× 10-31 kg and eV = 1.6 × 10-19 J) A) 26 B) 24 C) 42 D) 12
