A machine has an IMA of 6.0 and an AMA of 5.0. What is the work input needed by the machine to raise a 40 kg load by 3.0 m?

To find the work input needed by the machine, we need to use the formula:

Work Input (WI) = Output Force × Output Distance

In this case, the output force is the weight of the load, and the output distance is the distance the load is being raised.

1. Calculate the weight of the load:
Weight = Mass × Acceleration due to gravity
Weight = 40 kg × 9.8 m/s²
Weight = 392 N

2. Calculate the output force, which is the weight of the load:
Output Force = Weight = 392 N

3. Calculate the output distance, which is the distance the load is being raised (3.0 m in this case).

Now, to calculate the work input, we need to use the concept of the Ideal Mechanical Advantage (IMA) and the Actual Mechanical Advantage (AMA).

The formula for IMA is:

IMA = Output Distance / Input Distance

where Input Distance is the distance over which the input force is applied. In this case, the work input is the force applied to the machine times the input distance.

The formula for AMA is:

AMA = Output Force / Input Force

where Input Force is the force applied to the machine.

Given that IMA = 6.0 and AMA = 5.0, we can rearrange the formulas to solve for the input distance and input force:

Input Distance = Output Distance / IMA

Input Force = Output Force / AMA

4. Calculate the input distance:
Input Distance = Output Distance / IMA
Input Distance = 3.0 m / 6.0
Input Distance = 0.5 m

5. Calculate the input force:
Input Force = Output Force / AMA
Input Force = 392 N / 5.0
Input Force = 78.4 N

6. Calculate the work input:
Work Input = Input Force × Input Distance
Work Input = 78.4 N × 0.5 m
Work Input = 39.2 Joules

Therefore, the work input needed by the machine to raise a 40 kg load by 3.0 m is 39.2 Joules.