Three worked problems build intuition for uniform circular motion in vertical planes. The first finds speed and period for a ball circling inside a conical bowl using force balance; the second applies energy conservation to find speed at the top of a vertical loop; the third derives tension and period for a conical pendulum. Useful for students practicing how gravity, normal force, and tension combine to produce centripetal acceleration in different geometries.
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Scene 1: Worked ExampleOn-Screen Title: Worked Example: Motion in a Conical Bowl Question: A small ball of mass $m$ moves in uniform circular motion on the smooth inner surface of a conical bowl of radius $r$ with velocity $V$, making an angle $\theta$ with the horizontal. Find: (a) Speed $V$, (b) Period $T$. Answer:(a) Speed: $V = \sqrt{r g \tan\theta}$ (b) Period: $T = \frac{2\pi r}{V}$ Scene 2: Try It YourselfOn-Screen Title: Try It Yourself: Vertical Loop and Energy Conservation Question: On a smooth path, a vertical semicircle of radius $r$ meets an incline at lowest point A. A small ball of mass $m$ is released from rest at point P on the incline at height $h$ above A, moving to the highest point B. Find speed $V$ at highest point B in terms of $g$, $h$, and $r$. Answer:By conservation of energy: $mgh = mg(2r) + \frac{1}{2}mV^2$ Speed: $V = \sqrt{2g(h - 2r)}$ Scene 3: Try ItOn-Screen Title: Try It: Conical Pendulum Tension and Period Question: A string of length $l$ is fixed at the ceiling, and a small ball of mass $m$ tied to the other end undergoes uniform circular motion in a horizontal plane. The angle with the vertical is $\theta$. Find: (i) Tension in the string, (ii) Period of circular motion. Answer:(i) Tension = $\frac{mg}{\cos\theta}$ (ii) Period = $2\pi \sqrt{\frac{l \cos\theta}{g}}$ IMPORTANT LANGUAGE RULE: All on-screen text must be in Arabic EXCEPT mathematical equations, formulas, laws, symbols, variables, and numerical expressions. Keep every equation, formula, and law exactly as provided, without translating or changing it. Translate only the explanatory text, headings, labels, and descriptions into Arabic. Arabic text must use proper RTL direction and correct Arabic typography. Do not use English text anywhere else on the screen. Audience: student