MHTCET 22 Apr 2025 Shift 1 Paper

Concept:When a vehicle moves on a circular horizontal track, the suspended bob tilts because the pseudo force acting on it equals the centripetal force, and at equilibrium the ratio of horizontal to vertical forces gives $\tan \theta$.Explanation:Given: speed $v = 10 \, \mathrm{m/s}$, radius $R = 20 \, \mathrm{m}$, and $g = 10 \, \mathrm{m/s^2}$.The centripetal acceleration of the vehicle is:$$a_c = \frac{v^2}{R} = \frac{(10)^2}{20} = \frac{100}{20} = 5 \, \mathrm{m/s^2}$$This horizontal acceleration acts on the bob as well, so the string is inclined at an angle $\theta$ with the vertical.For equilibrium of the bob, the vertical component of tension balances the weight, and the horizontal component provides the required centripetal force.Dividing the horizontal equation by the vertical equation gives:$$\tan \theta = \frac{a_c}{g}$$Substituting the values:$$\tan \theta = \frac{5}{10} = 0.5$$Hence:$$\theta = \tan^{-1}(0.5)$$Answer:The angle made by the string with the vertical is $\tan^{-1}(0.5)$, so the correct option is Option A.