Ropes and Pulleys: Forces and Accelerations in Connected Systems

Chapter 5: Section 5.8 Ropes and Pulleys

Forces on Connected Masses with Pulleys

When two or more masses are connected by ropes and pulleys, the analysis of forces and accelerations requires careful application of Newton's laws. Pulleys are often assumed to be massless and frictionless, and ropes are considered massless and inextensible for introductory physics problems.

• Key Point 1: Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In ideal problems, the tension is the same throughout a massless rope.

• Key Point 2: Free-body diagrams are essential for analyzing each mass separately. Forces acting on each object include gravity, tension, and normal force (if the object is on a surface).

• Key Point 3: Newton's Second Law ($F = ma$) is applied to each mass. The acceleration of connected masses is related by the geometry of the rope and pulley system.

Example: Table and Hanging Masses

Consider Box A (weight 80 N) resting on a table, connected by a rope over a pulley to Box B (hanging, weight variable). The table exerts a normal force on Box A, which depends on the weight of Box B.

• Forces on Box A: Gravity (downward), normal force from table (upward), and tension from rope (upward).

• Forces on Box B: Gravity (downward), tension from rope (upward).

• Normal Force Calculation: The normal force $N$ on Box A is given by: $N = W_A - T$ where $W_A$ is the weight of Box A and $T$ is the tension in the rope.

• Tension Calculation: If the system is at rest or moving at constant velocity, $T = W_B$ (weight of Box B).

• Normal Force for Different Weights of Box B:

  • (a) $W_B = 55$ N: $N = 80 - 55 = 25$ N

  • (b) $W_B = 70$ N: $N = 80 - 70 = 10$ N

  • (c) $W_B = 90$ N: $N = 80 - 90 = -10$ N (Box A would lift off the table)

Additional info: If $N$ becomes negative, Box A is no longer in contact with the table and will be lifted by the tension in the rope.

Lifting a Stage Set: Two-Mass Pulley System

When two masses are connected by a rope over a pulley, their accelerations are related and the tension in the rope is the same for both masses. This is a classic Atwood machine scenario.

• Key Point 1: Draw separate free-body diagrams for each mass. Identify all forces: tension (upward), gravity (downward).

• Key Point 2: The acceleration of each mass is equal in magnitude but opposite in direction (one goes up, the other goes down).

• Key Point 3: The tension force is the same throughout the rope if the pulley is ideal (massless and frictionless).

Equations for the System

• Let $m_M$ be the mass of the stagehand, $m_S$ the mass of the set, $a_{MY}$ and $a_{SY}$ their accelerations. For the stagehand: $T - m_M g = m_M a_{MY}$ For the set: $T - m_S g = -m_S a_{MY}$ (since $a_{SY} = -a_{MY}$)

• Solving for $T$ from the first equation: $T = m_M a_{MY} + m_M g$ Substitute into the second equation: $m_M a_{MY} + m_M g - m_S g = -m_S a_{MY}$ Combine terms: $(m_M + m_S) a_{MY} = (m_S - m_M) g$ So, $a_{MY} = \frac{(m_S - m_M)g}{m_M + m_S}$

Example Calculation

• If $m_M = 100$ kg and $m_S = 200$ kg: $a_{MY} = \frac{(200 - 100) \times 9.8}{100 + 200} = \frac{100 \times 9.8}{300} = 3.27$ m/s2

• This is the acceleration with which the stagehand is lifted and the set falls.

Additional info: If the masses are equal, the acceleration is zero and the system is in equilibrium. If the set is much heavier, the acceleration approaches $g$ (free fall).

Summary Table: Forces and Accelerations in Pulley Systems

Object	Forces Acting	Equation	Acceleration
Mass on Table (A)	Gravity, Normal, Tension	$N = W_A - T$	Depends on $W_B$
Hanging Mass (B)	Gravity, Tension	$T = W_B$ (if static)	Downward if $W_B > W_A$
Stagehand (M)	Gravity, Tension	$T - m_M g = m_M a_{MY}$	$a_{MY} = \frac{(m_S - m_M)g}{m_M + m_S}$
Set (S)	Gravity, Tension	$T - m_S g = -m_S a_{MY}$	Opposite to $a_{MY}$

Additional info: These principles apply to many real-world systems, such as elevators, cranes, and mechanical hoists, where understanding the forces and accelerations is crucial for safety and design.