6.7 — Worked problems — advanced Newton’s-law applications

Seven worked problems exercising Chapter 6 tools.

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Problem 1 (§6.2) — max speed on a horizontal curve

Curve of radius 80 m with \mu_s = 0.6. Max speed without slipping?

Solution: v_\text{max} = \sqrt{\mu_s g r} \approx 21.7\ m/s \approx 78\ km/h.

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Problem 2 (§6.2) — banked turn angle

Road designed for 100 km/h on radius 500 m, no friction. Bank angle?

Solution: v = 27.8 m/s. \tan\theta = v^2/(gr) = 0.158. \theta \approx 9°.

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Problem 3 (§6.2) — min speed at top of vertical loop

Roller coaster with loop radius 5 m. Minimum speed at top?

Solution: v_\text{min} = \sqrt{gr} = 7\ m/s.

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Problem 4 (§6.3) — cyclist decelerating in a curve

Curve 40 m, speed 12 m/s, tangential deceleration -1.5 m/s². Total acceleration?

Solution:

• a_r = 3.6\ m/s^2
• a_t = -1.5\ m/s^2
• a \approx 3.9\ m/s^2

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Problem 5 (§6.5) — terminal velocity of a small ball

Small ball diameter 1 cm (A \approx 8 \times 10^{-5} m²), mass 10^{-3} kg, C_d = 0.5, \rho = 1.2. Terminal velocity?

Solution: v_t = \sqrt{2mg/(\rho C_d A)} \approx 20\ m/s.

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Problem 6 (§6.4) — accelerating frame

Elevator accelerating up at 2 m/s². A pendulum hanging inside. Steady-state angle?

Solution: For vertical acceleration, the pendulum hangs straight (angle 0). If the acceleration were horizontal (say 2 m/s² to the right), the pendulum would tilt back: \tan\theta = a_0/g \Rightarrow \theta \approx 11.5°.

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Problem 7 (§6.2 + §6.6) — Earth-orbiting satellite

Satellite at 600 km altitude. Orbital speed? (R_E = 6371 km, GM = 4 \times 10^{14})

Solution: r = 6971 km. v = \sqrt{GM/r} \approx 7580\ m/s.

Period: T \approx 5780 s \approx 96 min.

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Preview of §6.8

Twenty practice problems with final answers.

📚 See also: Halliday Vol 1, Ch 6 — Worked Examples.

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