Question

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:

Step 1: Analyze the forces on the load

The load has a mass \( m = 10\space kg \). The gravitational force is \( F_g=mg \), where \( g = 9.8\space m/s^2 \), so \( F_g=10\times9.8 = 98\space N \). Let the applied force be \( F \), and the net force on the load is \( F_{net}=ma \), where \( a \) is between \( 1\space m/s^2 \) and \( 2\space m/s^2 \). The tension in the string (related to the pulley system) should satisfy \( T - mg=ma \) (if the pulley system is lifting the load, tension \( T \) is related to the applied force by the number of supporting ropes, say \( n \), so \( T = nF \)). So \( nF - mg=ma \), \( nF=m(g + a) \).

Step 2: Calculate the required number of supporting ropes (\( n \))

For \( a = 1\space m/s^2 \), \( m(g + a)=10\times(9.8 + 1)=108\space N \). We need \( nF<108\space N \) and \( F < 35\space N \). Let's solve for \( n \): \( n>\frac{108}{35}\approx3.09 \). For \( a = 2\space m/s^2 \), \( m(g + a)=10\times(9.8+2) = 118\space N \), \( n>\frac{118}{35}\approx3.37 \). Since \( n \) must be an integer (number of pulleys/supporting ropes), and we have 3 - 5 pulleys, let's check with \( n = 4 \). Then \( F=\frac{m(g + a)}{n} \). For \( a = 1\space m/s^2 \), \( F=\frac{108}{4}=27\space N<35\space N \). For \( a = 2\space m/s^2 \), \( F=\frac{118}{4} = 29.5\space N<35\space N \). So a pulley system with 4 supporting ropes (using 3 - 5 pulleys, e.g., 2 fixed and 2 movable pulleys or other combinations to get 4 supporting ropes) would work. The design would have the load attached to a system where 4 segments of the string support it, the applied force is applied to the free end, the pulleys are attached to the beam (fixed pulleys) and the load (movable pulleys) as needed, with massless string, 3 - 5 pulleys (e.g., 2 fixed, 2 movable, total 4 pulleys), and the wooden plank can be used if needed for mounting, but since it's 0 - 1, we can choose to use or not. The applied force \( F \) would be less than 35 N (e.g., 27 - 29.5 N) and the acceleration would be between 1 and 2 \( m/s^2 \).

Answer:

A pulley system with 4 supporting ropes (using 3 - 5 pulleys, e.g., 2 fixed and 2 movable pulleys) can be designed. The applied force \( F \) (less than 35 N) and the number of pulleys are arranged such that \( n = 4 \) (number of supporting ropes), so \( F=\frac{m(g + a)}{n} \) satisfies \( 1\space m/s^2\leq a\leq2\space m/s^2 \) and \( F < 35\space N \). The pulley system consists of the 10 - kg load, massless string, 4 pulleys (e.g., 2 fixed on the beam, 2 movable attached to the load), and optional wooden plank (not needed here) with hooks attached as needed.