A pendulum consists of a bob of mass 'm' suspended by a...

A pendulum consists of a bob of mass 'm' suspended by a light, inextensible string of length $' \ell^{\circ}$ fixed at $\mathrm{O}$. What is the horizontal velocity ' $\mathrm{V}^{3}$ to be given to the bob at position $\mathrm{A}$, so that it passes position B but does not reach position $\mathrm{C} ?$ (a) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<2 \sqrt{\mathrm{g} \ell}$ (b) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{2 \mathrm{~g} \ell}$ (c) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$ (d) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$

A pendulum consists of a bob of mass 'm' suspended by a light, inextensible string of length $' \ell^{\circ}$ fixed at $\mathrm{O}$. What is the horizontal velocity ' $\mathrm{V}^{3}$ to be given to the bob at position $\mathrm{A}$, so that it passes position B but does not reach position $\mathrm{C} ?$
(a) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<2 \sqrt{\mathrm{g} \ell}$
(b) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{2 \mathrm{~g} \ell}$
(c) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$
(d) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$

Key Concepts

Pendulum Dynamics in Varying Regions

Determining whether a pendulum reaches a particular point on its path involves analyzing the dynamics of its motion. This includes assessing how the initial horizontal velocity influences the bob's trajectory, the interplay of energy conversion, and the gravitational potential differences as the bob travels along the circular arc. Such analysis typically reveals the range of acceptable speeds to achieve specific motion outcomes.

Conservation of Energy

In problems involving pendulum motion, conservation of energy is a fundamental principle used to relate the kinetic and potential energy at different positions along the path. As the pendulum swings, its gravitational potential energy is converted into kinetic energy and vice versa. This principle allows us to determine the changes in speed by comparing the energies at various points in the motion.

Centripetal Force and Circular Motion

A pendulum bob undergoes circular motion, so centripetal force, which keeps the bob moving along its curved path, must be provided by the tension in the string and the gravitational force. Understanding the requirements for centripetal force is crucial, particularly when determining the critical speed at which the only available forces are sufficient to maintain the circular trajectory without the string going slack.

Critical Velocity for Maintaining Tension

For a pendulum to complete a circular path or just swing to certain points without reaching others, there are minimum velocity requirements. At the highest point of the motion, the tension in the string must be non?negative, which imposes a minimum speed. Analyzing these boundary conditions helps determine the range of initial velocities needed to ensure that the bob reaches some positions but not others.

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