In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is $$T = 2\pi \sqrt {{{7\left( {R - r} \right)} \over {5g}}} $$. The values of R and r are measured to be (60 $$ \pm $$ 1) mm and (10 $$ \pm $$ 1) mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x₀. Consider two cases:
(i) when the block is at x₀; and
(ii) when the block is at x = x₀ + A.
In both cases, a particle with mass m( < M) is softly placed on the block after which they stick on each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure P_i = 10⁵ Pa and volume V_i = 10^-3 m³ changes to a final state at P_f = $$\left( {{1 \over {32}}} \right) \times {10^5}\,Pa$$ and V_f = 8 $$ \times $$ 10^-3 m³ in an adiabatic quasi-static process, such that P³V⁵ = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at P_i followed by an isochoric (isovolumetric) process at volume V_f. The amount of heat supplied to the system in the two-step process is approximately

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10^oC. Now the end P is maintained at 10^oC, while the end S is heated and maintained at 400^oC. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is 1.2 $$ \times $$ 10^-5 K^-1 , the change in length of the wire PQ is