Question

directions: create a model to help you solve the problem below. all models must include an explanation. problem: design a pulley system attached to a sturdy beam that lifts a 10-kg load with an acceleration between 1 m/s² and 2 m/s² with an applied force of less than 35 n. the pulley system must be composed of the following: - (1) 10-kg load - massless string - 3 to 5 pulleys - 0 to 1 wooden plank of negligible mass assume that you are able to attach as many hooks as needed to the beam and the wooden plank.

Explanation:

Step1: Analyze the force on the load

The load has a mass \( m = 10\space kg \). Let the acceleration be \( a \) (where \( 1\space m/s^{2}\leq a\leq2\space m/s^{2} \)) and the tension in the string lifting the load be \( T \). Using Newton's second law \( F = ma \), the net force on the load is \( T - mg= ma \) (taking upward as positive), so \( T=m(g + a) \).

For \( a = 1\space m/s^{2} \), \( T=10\times(9.8 + 1)=108\space N \)

For \( a = 2\space m/s^{2} \), \( T = 10\times(9.8+ 2)=118\space N \)

Step2: Relate tension to applied force in pulley system

In a pulley system with \( n \) supporting strands (the number of strands of string supporting the load), the applied force \( F \) is related to the tension \( T \) by \( F=\frac{T}{n} \) (assuming ideal pulleys, massless string, etc.). We know that \( F< 35\space N \), so \( n>\frac{T}{35} \)

For the minimum \( T = 108\space N \), \( n>\frac{108}{35}\approx3.09 \)

For the maximum \( T = 118\space N \), \( n>\frac{118}{35}\approx3.37 \)

Since \( n \) must be an integer, \( n = 4 \) (because \( n = 3 \) would give \( F=\frac{108}{3}=36\space N>35\space N \) which is not allowed, and \( n = 4 \) gives \( F=\frac{108}{4} = 27\space N<35\space N \) and \( F=\frac{118}{4}=29.5\space N<35\space N \))

Step3: Design the pulley system

We can use 4 supporting strands. A possible design: Use 2 fixed pulleys and 2 movable pulleys. Attach the load to the movable pulleys. The fixed pulleys are attached to the beam. The string is threaded through the pulleys such that 4 strands support the load. The applied force is applied to the free end of the string.

Answer:

To design the pulley system:

1. Components: 2 fixed pulleys, 2 movable pulleys, massless string, 10 - kg load, and (optional) a wooden plank (not necessary for the force calculation but can be used for attachment).
2. String Threading: Thread the string through the pulleys so that there are 4 strands of string supporting the load (the movable pulleys). Attach the fixed pulleys to the beam. Attach the load to the movable pulleys. Apply the force to the free - end of the string.
3. Force and Acceleration Check: With \( n = 4 \) supporting strands, the applied force \( F=\frac{T}{4} \). For the load, \( T=m(g + a) \). When \( a = 1\space m/s^{2} \), \( T = 108\space N \), \( F = 27\space N<35\space N \); when \( a=2\space m/s^{2} \), \( T = 118\space N \), \( F = 29.5\space N<35\space N \), which satisfies the requirements.