A worked-example animation solving three uniform circular motion problems: computing period from revolutions per time, converting to angular and linear speed, finding centripetal acceleration from angular speed, and using a spring's stretch to determine elastic force, speed, and angular speed. Each step shows the formula, substituted values, and final numeric answer with units. Useful for students practicing circular motion calculations and teachers reviewing problem-solving technique for period, speed, and centripetal force relationships.
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Scene 4On-Screen Title: Problem 4: Period, Angular and Linear SpeedsQuestion: (a) An object undergoes uniform circular motion, completing 10 revolutions in $2.0\text{ s}$ on a circle of diameter $0.40\text{ m}$. Find the period in seconds. (b) Find its angular speed in $\text{rad/s}$ and linear speed in $\text{m/s}$. Answer:(a) Period ($T$): $T = \frac{2.0}{10} = \mathbf{0.20\text{ s}}$ (b) Angular speed ($\omega$): $\omega = \frac{2\pi}{T} = \frac{2\pi}{0.20} \approx \mathbf{31.42\text{ rad/s}}$; Linear speed ($v$): $v = \omega r = 31.42 \times 0.20 = \mathbf{6.28\text{ m/s}}$ Scene 5On-Screen Title: Problem 5: Centripetal Acceleration with Angular SpeedQuestion: A small ball of mass $0.50\text{ kg}$ is tied to a string of length $2.0\text{ m}$ undergoing uniform circular motion on a smooth horizontal surface with an angular speed of $3.0\text{ rad/s}$. Find its centripetal acceleration. Answer:Formula: $a_c = \omega^2 r$ Calculation: $a_c = (3.0)^2 \times 2.0 = 9 \times 2.0 = \mathbf{18\text{ m/s}^2}$ Scene 6On-Screen Title: Problem 6: Spring Force, Velocity, and Angular SpeedQuestion: A small ball of mass $8.0 \times 10^{-2}\text{ kg}$ is attached to a spring with initial length $0.13\text{ m}$ and spring constant $60\text{ N/m}$, undergoing uniform circular motion to a spring length of $0.15\text{ m}$. Find to 2 significant figures: (i) Elastic force, (ii) Speed, (iii) Angular speed. Answer:(i) Elastic force: $F = k(L - L_0) = 60(0.15 - 0.13) = \mathbf{1.2\text{ N}}$ (ii) Speed: $v = \sqrt{\frac{F \cdot r}{m}} = \sqrt{\frac{1.2 \times 0.15}{8.0 \times 10^{-2}}} = \mathbf{1.5\text{ m/s}}$ (iii) Angular speed: $\omega = \frac{v}{r} = \frac{1.5}{0.15} = \mathbf{10\text{ rad/s}}$ 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