5.7 — Applications — pulleys, inclined planes, elevators

Classic systems that appear in every exam and real-world problem.

Elevator — force in an accelerated frame

A 70 kg person on a scale in an elevator, accelerating upward at 2 m/s². What does the scale read?

FBD of the person:

• Gravity: -mg = -686 N
• Normal (scale): +N

Second law: N - mg = ma \Rightarrow N = m(g + a) = 70 \cdot 11.8 = 826\ N

Scale reads 826 N — which (dividing by g) reads as 84 kg. The person feels "heavier".

Three cases:

• Elevator accelerating up: N > mg (feels heavier)
• Elevator accelerating down: N < mg (feels lighter)
• Free fall (a = -g): N = 0 (weightless)

Simple pulley — two hanging masses

Two masses m_1 = 3 kg and m_2 = 5 kg hang from a frictionless pulley. Acceleration and tension?

FBD of m_1 (lighter, rises):

• T - m_1 g = m_1 a

FBD of m_2 (heavier, falls):

• m_2 g - T = m_2 a

Add:

\[ (m_2 - m_1)g = (m_1 + m_2)a \Rightarrow a = \frac{(m_2 - m_1)g}{m_1 + m_2} = 2.45\ m/s^2 \]

Tension:

\[ T = m_1(g + a) = 36.75\ N \]

Check: T should lie between m_1 g = 29.4 and m_2 g = 49\ N. ✓

Inclined plane with friction

A 10 kg box on a 25° slope with \mu_k = 0.2. Acceleration?

Decompose weight:

• Along slope: mg\sin 25° = 41.4\ N (down-slope)
• Perpendicular: mg\cos 25° = 88.8\ N

Normal: N = 88.8\ N

Kinetic friction: f_k = \mu_k N = 17.8\ N (up-slope, opposing motion)

Net along slope: 41.4 - 17.8 = 23.6\ N

Acceleration: a = 2.36\ m/s^2 (down-slope)

Two connected boxes on a horizontal surface

Two boxes m_1 = 2 kg and m_2 = 3 kg connected by a rope. A 20 N force pulls m_2. Acceleration and tension?

Whole system: F = (m_1 + m_2) a \Rightarrow a = 4\ m/s^2

Just m_1: the rope applies T. T = m_1 a = 8\ N.

Check on m_2: F - T = m_2 a \Rightarrow 20 - 8 = 12 = 3 \cdot 4 ✓

Problems with an angled force

A 50 kg box pulled by 200 N at 30° above horizontal. Horizontal surface with \mu_k = 0.15.

Decompose force:

• F_x = 200\cos 30° = 173.2\ N
• F_y = 200\sin 30° = 100\ N (upward)

Normal: weight - F_y (the pull lifts a bit) = 490 - 100 = 390\ N

Friction: f_k = 0.15 \cdot 390 = 58.5\ N

Acceleration: a = (173.2 - 58.5)/50 = 2.29\ m/s^2

Note: an upward angled force lightens the normal force — so friction is smaller.

A few notes and common mistakes

1. Draw an FBD for each object. Multi-object systems need multiple FBDs.

2. Tension is uniform in a massless pulley. If the pulley had mass, tensions on either side would differ.

3. Get the direction right. If you assume the wrong direction for acceleration, the formulas give a negative answer — the algebra fixes signs.

What you should be able to do

• Elevator problems — normal force vs frame acceleration
• Simple pulley — two hanging bodies, a and T
• Inclined plane with friction — combining friction and weight decomposition
• Connected boxes — whole-system behavior and internal tension
• Angled force — how it modifies normal force

Preview of §5.8

Worked problems exercising all these tools in varied scenarios.

📚 See also: Halliday Vol 1, Ch 5, §5.7-5.9.

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