5.8 — Worked problems — Newton’s laws in action

Seven worked problems exercising all Chapter 5 tools.

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Problem 1 (§5.3) — two forces at angles

Two forces on a 2 kg object: \vec F_1 = 10\hat i + 6\hat j\ N, \vec F_2 = -4\hat i + 3\hat j\ N. Acceleration?

Solution: \vec F_\text{net} = 6\hat i + 9\hat j. \vec a = 3\hat i + 4.5\hat j\ m/s^2. Magnitude \approx 5.4\ m/s^2.

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Problem 2 (§5.6) — friction coefficient

A box on a slope starts sliding at \theta = 27°. Find \mu_s.

Solution: \mu_s = \tan 27° \approx 0.510.

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Problem 3 (§5.5) — weight on the Moon

An astronaut of mass 80 kg. Weight on Earth, Moon, Mars?

Solution:

• Earth: 784\ N
• Moon: 130\ N
• Mars: 296\ N

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Problem 4 (§5.7) — Atwood machine

Two masses 4 kg and 6 kg. Acceleration? Tension?

Solution:

• a = 1.96\ m/s^2
• T = 47\ N

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Problem 5 (§5.3 + §5.6) — car braking

A 1000 kg car at 20 m/s on wet road (\mu_k = 0.5). Stopping distance?

Solution:

• f_k = 4900\ N
• a = -4.9\ m/s^2
• Kinematics: d = v^2/(2|a|) = 40.8\ m

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Problem 6 (§5.7) — train and car

Engine 1500 kg pulls a car 500 kg. Engine force 6000 N, total friction 1000 N. Acceleration? Coupler tension?

Solution:

• Whole system: a = 2.5\ m/s^2
• Just the car: T - f_\text{car} = m_\text{car} a. Assume friction ∝ mass: f_\text{car} = 250\ N. T = 1500\ N.

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Problem 7 (§5.4) — two ice skaters

Skaters at 70 kg and 50 kg standing on ice. The first pushes the second with 120 N. Acceleration of each?

Solution:

• Second skater: 120/50 = 2.4\ m/s^2
• Third law: the second exerts -120 N on the first.
• First skater: 120/70 = 1.71\ m/s^2 backward.

Note: they move in opposite directions — momentum conservation (Ch 9).

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Preview of §5.9

Twenty practice problems.

📚 See also: Halliday Vol 1, Ch 5 — Worked Examples.

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