Under less than ideal conditions, a student pushes a 6 kg cart holding 2 boxes of books up a ramp to a platform that is 1.8 m high in 42 sec. Each box contains 12 kg of books. The ramp is inclined at an angle of 15 degrees. He pushes w/ a steady force of 70 N to move it up.
Find: -Work in
-Work out
-IMA
-AMA
-Efficiency
-Power
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To find the various quantities, we need to follow a step-by-step approach:
Step 1: Calculate the work input (Wi):
The work input, also known as the work done by the student in pushing the cart up the ramp, can be calculated using the formula:
Wi = Force × Distance
Given that the force exerted by the student is 70 N and the distance covered is the inclined height of the ramp, which is 1.8 m, we can calculate Wi:
Wi = 70 N × 1.8 m
Step 2: Calculate the work output (Wo):
The work output, which represents the work done against gravity when lifting the boxes, can be calculated as the sum of the work done on each box individually. The work done on each box is given by:
(Force × Distance) = (Weight of the box × Distance)
For each box, the weight is determined by the mass and acceleration due to gravity (9.8 m/s²). Therefore, the weight can be calculated as:
Weight = mass × acceleration due to gravity
For each box, the distance covered vertically is the same as the height of the ramp, which is 1.8 m. So, we can calculate Wo for each box:
Wo1 = (Weight of Box 1 × Distance)
Wo1 = (12 kg × 9.8 m/s² × 1.8 m)
Similarly, for the second box:
Wo2 = (Weight of Box 2 × Distance)
Wo2 = (12 kg × 9.8 m/s² × 1.8 m)
Finally, we find the total work output, Wo, by summing the work done on each box:
Wo = Wo1 + Wo2
Step 3: Calculate the Ideal Mechanical Advantage (IMA):
The IMA is the mechanical advantage calculated based on the physical dimensions of the inclined plane (ramp). It is given by the formula:
IMA = Length of the ramp / Height of the ramp
Given that the angle of inclination is 15 degrees, we can calculate the height and length of the ramp using trigonometry. Let's assign the height as h and the length as L:
h = height of the ramp = 1.8 m
L = length of the ramp
Using the trigonometric relation:
sin(15 degrees) = h / L
We can rearrange the equation to solve for L:
L = h / sin(15 degrees)
Step 4: Calculate the Actual Mechanical Advantage (AMA):
The AMA is the mechanical advantage calculated based on the force applied and the force required. It is given by the formula:
AMA = Output force / Input force
In this case, the output force is the force required to lift the boxes against gravity, which is the weight:
Output force = Weight of the boxes
The input force is the force exerted by the student, which is 70 N.
Since we have already calculated the weight of each box (Weight1 and Weight2), we can calculate the AMA for each box:
AMA1 = Weight1 / Input force
AMA2 = Weight2 / Input force
Finally, we find the total AMA by summing the AMAs for both boxes:
AMA = AMA1 + AMA2
Step 5: Calculate the Efficiency:
Efficiency is defined as the ratio of the work output to the work input, expressed as a percentage:
Efficiency = (Wo / Wi) × 100
Step 6: Calculate the Power:
Power is defined as the rate at which work is done or energy is transferred. It can be calculated using the formula:
Power = Work / Time
Since we have found the work input (Wi), we can calculate the power:
Power = Wi / Time
Given the time is 42 seconds, we can calculate the various quantities using the above steps.