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a)  2.29 u00d7 107 Hz b) 2.70 u00d7 10-12 J Radius, \gamma=0.390 \mathrm{~m}magnetic fie k, B=1.50 \mathrm{~T}Mass of proton, m_{p}=1.67 \times 10^{-27} \mathrm{~kg}charge of proton, q_{p}=1.60 \times 10^{-19} \mathrm{C}(a) Centripetal force is provited by Magnetic forc. de, F_{c}=F_{B}\Rightarrow \frac{m_{p} v^{2}}{\gamma}=q_{p} v BOscillator frequency, F=\frac{V}{2 \pi r}-2\begin{array}{l}\Rightarrow \Rightarrow \frac{V}{\gamma}=\frac{q_{p} B}{m_{p}} \therefore \frac{V}{2 \pi \gamma}=\frac{q_{p} B}{2 \pi m_{p}} \\\therefore F=\frac{V}{2 \pi_{\gamma}}=\frac{q_{p} B}{2 \pi m_{p}}=\frac{1.6 \times 10^{-19}(\times 1.50 T}{2 \pi \times 1.67 \times 10^{-27} \mathrm{~kg}} \\\therefore F=2.29 \times 10 \mathrm{~Hz}\end{array}(b) speed, V=F \times 2 \pi r=2.29 \times 10^{7} H z \times 2 \pi \times 0.39 \mathrm{~m} V=5.61 \times 10^{7} \mathrm{~m} / \mathrm{s}\therefore Kinctic energy, K E=\frac{1}{2} m_{p} v^{2}=\frac{1}{2}Here, speed Is comparable to speed of light \left(c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right).\therefore kinetic endrg], k=(r-1) m c^{2}where; r=\frac{1}{\sqrt{1-(v / c)^{2}}}=1 / \frac{1}{\sqrt{1-\left(\frac{5.61 \times 10 \% / 3}{3 \times 10^{3} \mathrm{~m} / \mathrm{s}}\right)^{2}}}=1.018\begin{aligned}\therefore K & =(1.018-1) \times 1.67 \times 10^{-27} \mathrm{~kg} \times\left(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)^{2} \\& =2.70 \times 10^{-12} \mathrm{~J}\end{aligned} ...