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 forces on the load

The load has mass \( m = 10\,\text{kg} \). Let the tension in the string supporting the load be \( T \). Using Newton's second law \( F_{\text{net}}=ma \), the net force on the load is \( T - mg=ma \) (upward tension, downward weight). So \( T = m(g + a) \). We need to consider the range of \( a \) (1 to \( 2\,\text{m/s}^2 \)) and \( g = 9.8\,\text{m/s}^2 \).

For \( a = 1\,\text{m/s}^2 \): \( T_1=10\times(9.8 + 1)=108\,\text{N} \)

For \( a = 2\,\text{m/s}^2 \): \( T_2=10\times(9.8 + 2)=118\,\text{N} \)

Step2: Relate tension to applied force (pulley system)

In a pulley system with \( n \) supporting strands (mechanical advantage \( MA = n \)), the applied force \( F=\frac{T}{n} \) (since tension is distributed among \( n \) strands, assuming ideal pulleys, massless string). We need \( F<35\,\text{N} \), so \( n>\frac{T}{35} \).

For \( T = 108\,\text{N} \): \( n_1>\frac{108}{35}\approx3.09 \), so \( n\geq4 \)

For \( T = 118\,\text{N} \): \( n_2>\frac{118}{35}\approx3.37 \), so \( n\geq4 \)

Step3: Design the pulley system

We can use 4 supporting strands (so \( n = 4 \)). A combination of 2 fixed and 2 movable pulleys (total 4 pulleys, within 3 - 5 range) can give \( n = 4 \). Attach the fixed pulleys to the beam, movable pulleys to the load (or a plank holding the load). The string is threaded through fixed and movable pulleys, with the applied force at the free end.

Check applied force: For \( T = 118\,\text{N} \), \( F=\frac{118}{4}=29.5\,\text{N}<35\,\text{N} \), which works. For \( T = 108\,\text{N} \), \( F=\frac{108}{4}=27\,\text{N}<35\,\text{N} \), also works.

Step4: Construct the model

• Components: 2 fixed pulleys (attached to the beam), 2 movable pulleys (attached to the load or a plank holding the load), massless string, 10 - kg load.
• Assembly: Thread the string through a fixed pulley, then a movable pulley, then a fixed pulley, then a movable pulley. Attach the load to the movable pulleys. Apply force at the free end of the string. The number of supporting strands \( n = 4 \), so mechanical advantage is 4, reducing the force needed to lift the load with the required acceleration.

Answer:

A pulley system with 2 fixed pulleys, 2 movable pulleys (total 4 pulleys, within 3 - 5), massless string, and the 10 - kg load. The string is threaded to have 4 supporting strands (mechanical advantage 4), allowing an applied force \( F<35\,\text{N} \) to lift the load with acceleration \( 1 - 2\,\text{m/s}^2 \). The fixed pulleys attach to the beam, movable pulleys (or a plank with the load) are supported by the string, and force is applied at the string's free end.