A system of frictionless and massless pulleys is being used to pull up a load of 2520 N. What is the minimum force (in Newtons) required at the end of the rope (shown in dashed lines)?
Show Hint
When calculating the mechanical advantage of a pulley system, only count the segments of the rope that support the movable pulley(s). The rope segment you pull on (the effort rope) does not count towards the supporting segments unless it is pulling upwards on the load.
Step 1: Understanding the Concept:
This problem involves a pulley system, which is a simple machine used to reduce the effort required to lift a heavy load. For an ideal system (with frictionless and massless pulleys), the reduction in force is determined by the Mechanical Advantage (MA). The MA is equal to the number of rope segments that are actively supporting the load.
Step 2: Key Formula or Approach:
The relationship between load, effort, and mechanical advantage is: \
\[ \text{Mechanical Advantage (MA)} = \frac{\text{Load}}{\text{Effort}} \] \
For this pulley system, the MA is the number of rope segments supporting the movable pulleys to which the load is attached. Let this number be \(n\). The effort is the force \(F\) we need to find. \
\[ n = \frac{\text{Load}}{F} \implies F = \frac{\text{Load}}{n} \] \
Step 3: Detailed Explanation: 1. Identify the Load:
The load to be lifted is given as 2520 N. 2. Determine the Mechanical Advantage (n):
We need to count the number of rope segments pulling upwards on the lower block of pulleys that holds the load. By carefully examining the diagram:
There are three pulleys in the lower movable block.
Each of these three pulleys has two rope segments pulling upwards on it.
Counting these segments from left to right, we find there are 6 strands of rope directly supporting the 2520 N load. \
Therefore, the mechanical advantage \(n = 6\). 3. Calculate the Minimum Force (F):
Using the formula from Step 2:
\[ F = \frac{\text{Load}}{n} = \frac{2520 \text{ N}}{6} \]
\[ F = 420 \text{ N} \]
Step 4: Final Answer:
The minimum force required at the end of the rope is 420 N.
