Question
Establish the formula for work in the adiabatic process.

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A particle of charge $2.0 \times 10^{-8} \mathrm{C}$ and mass $2.0 \times 10^{-10} \mathrm{~g}$ is projected with a speed of $2.0 \times 10^3 \mathrm{~m} / \mathrm{s}^{-1}$ in a region with a uniform magnetic field of 0.10 T . The velocity is perpendicular to the field. Find the radius of the circle formed by the particle and also the time period.
A cord of negligible mass is wound round the rim of a fly wheel of mass $20 \ kg$ and radius $20 \ cm$. A steady pull of $25 N$ is applied on the cord as shown in Fig. $6.31$. The flywheel is mounted on a horizontal axle with frictionless bearings.
$(a)$ Compute the angular acceleration of the wheel.
$(b)$ Find the work done by the pull, when $2m$ of the cord is unwound.
$(c)$ Find also the kinetic energy of the wheel at this point. Assume that the wheel starts from rest.
$(d)$ Compare answers to parts $(b)$ and $(c).$
A $14.5kg$ mass, fastened to the end of a steel wire of unstretched length $1.0m$, is whirled in a vertical circle with an angular velocity of $2rev/s$ at the bottom of the circle. The cross-sectional area of the wire is $0.065cm^2$. Calculate the elongation of the wire when the mass is at the lowest point of its path.
A sample contains a mixture of $^{108}Ag$ and $^{110}Ag$ isotopes each having an activity of $8.0 \times 10^8$ disintegration per second. $^{110}Ag$ is known to have larger half-life than $^{108}Ag$. The activity A is measured as a function of time and the following data are obtained.
Time (s) Activity (A) ($10^8$ disinte- grations $s^{-1}$) Time (s) Activity (A) ($10^8$ disinte-grations $s^{-1}$)
20 11.799 200 3.0828
40 9.1680 300 1.8899
60 7.4492 400 1.1671
80 6.2684 500 0.7212
100 5.4115    
  1. Plot ln $\Big(\frac{\text{A}}{\text{A}_0}\Big)$ versus time.
  2. See that for large values of time, the plot is nearly linear. Deduce the half-life of $^{110}Ag$ from this portion of the plot.
  3. Use the half-life of $^{110}Ag$ to calculate the activity corresponding to $^{108}Ag$ in the first 50s.
  4. Plot In $\Big(\frac{\text{A}}{\text{A}_0}\Big)$ versus time for $^{108}Ag$ for the first 50s.
  5. Find the half-life of $^{108}Ag$.
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