6_Pulley_Problems_720

Overview of Pulley Problems

• These problems often arise in AP Physics and undergraduate physics courses.

• Key assumptions:

  • The mass of the pulley is negligible compared to that of the blocks.

  • The mass and weight of the rope are also considered negligible.

• Focus: Calculate acceleration and tension for various pulley systems.

Problem 1: Standard Pulley System

• Setup: A large block (M) connected to a small block (m) with a pulley.

• Assumptions: No friction between the surface and the large block.

• Analysis:

  • Free Body Diagram for both blocks:

    • Large block (M): Weight (Mg), Normal force (N) upward, tension (T) acts horizontally.

    • Small block (m): Weight (mg) downward, tension (T) upward.

  • Key Equations:

    • For the large block: T = Ma (1)

    • For the small block: mg - T = ma (2)

  • Acceleration:

    • Combine equations (1) and (2) to eliminate T: mg = (M + m)a.

    • Hence, a = (mg) / (M + m).

  • Solve with values (M=10 kg, m=1 kg):

    • a = 0.89 m/s².

  • Tension:

    • Use T = Ma:

    • T = 10 × 0.89 = 8.9 N.

Problem 2: Hanging Masses

• Setup: Two hanging blocks connected by a single rope.

• Assumptions: Same as previous problem.

• Analysis:

  • Free Body Diagram:

    • Large block (M): mg downward, tension upward.

    • Small block (m): T upward, mg downward.

  • The system uses Newton's second law:

    • For M: mg - T = Ma (3)

    • For m: T - mg = ma (4)

  • Combine (3) + (4) to find acceleration:

    • a = (Mg - mg) / (M + m).

  • Substitute values, M=10kg, m=1kg: a ≈ 8 m/s².

  • Solve for Tension:

    • Work from either equation: T = m(g + a) = 1(9.8 + 8) = 17.8 N.

Problem 3: Block on a Slope

• Setup: A block on a slope connected to another block hanging.

• Assumptions: No friction, and pulley mass is negligible.

• Force Analysis:

  • Components of weight for block on slope:

    • Weight (Mg) acting down, tension (T) acting up the slope.

    • Weight component down slope = mg sin θ.

    • Normal force = mg cos θ.

  • Newton's laws:

    • For block on slope: -mg sin θ + T = Ma (5)

    • For hanging block: T - mg = ma (6)

  • Combine equations to eliminate T:**

    • mg sin θ = (M + m)a

  • Substitute values for calculations: a ≈ 3.6 m/s²,

  • Tension calculation:

    • From equation, T = m(g + a) = 1(3.6 + 9.8) = 13.4 N.

Problem 4: Multiple Pulley System

• Setup: Block connected through multiple (6) pulleys.

• Force Analysis:

  • For constant speed lowering: Force = weight / number of strings.

  • For upward acceleration, use adjusted equation for F:

    • Sum all upward forces and relate to mg + ma.

• Example calculations for forces show benefits of using pulleys for large weights, e.g., lift a 300 kg block more easily.

General Tips for Solving Pulley Problems

• Draw Free Body Diagrams for visualizing forces.

• Identify directions of acceleration, choosing a consistent reference frame.

• Apply Newton's laws simultaneously to both blocks where necessary.

• Eliminate unknowns through clever addition or substitution.

• Keep track of signs for tension: generally positive for upwards and negative for downwards.

• Practice with different arrangements to solidify understanding.