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
To find the work in, we need to calculate the force exerted multiplied by the distance moved in the direction of the force. In this case, the student pushes the cart up the ramp, so the distance moved in the direction of the force is the height of the ramp, 1.8 m. The force exerted is given as 70 N. Therefore, the work in is calculated as:
Work in = Force × Distance
Work in = 70 N × 1.8 m
Work in = 126 J
Next, let's find the work out. Since the cart is moving up the ramp against the force of gravity, work out is equal to the weight of the cart and the boxes multiplied by the vertical distance moved. The weight is the mass multiplied by the acceleration due to gravity, which is 9.8 m/s^2. The total mass is the sum of the cart and the two boxes of books:
Mass of cart = 6 kg
Mass of each box = 12 kg
Total mass = Mass of cart + (2 × Mass of each box)
Total mass = 6 kg + (2 × 12 kg)
Total mass = 30 kg
The vertical distance moved is the height of the ramp, which is 1.8 m. Therefore, the work out is calculated as:
Work out = Weight × Distance
Work out = (Mass × Gravity) × Distance
Work out = (30 kg × 9.8 m/s^2) × 1.8 m
Work out = 529.2 J
The next parameter is the IMA (Ideal Mechanical Advantage). IMA is the ratio of the distance the output force moves to the distance the input force moves. In this case, the IMA is related to the angle of the ramp. The formula to calculate IMA for an inclined plane is:
IMA = Length of ramp / Height of ramp
Given that the ramp is inclined at an angle of 15 degrees, we can use trigonometry to find the height and length of the ramp. Let's assume the ramp length (hypotenuse) is L:
Height = L × sin(angle)
Height = L × sin(15 degrees)
Height = L × 0.259
Since we know the height is 1.8 m, we can solve for L:
1.8 = L × 0.259
L = 1.8 / 0.259
L ≈ 6.95 m
Therefore, the IMA is:
IMA = Length of ramp / Height of ramp
IMA = 6.95 m / 1.8 m
IMA ≈ 3.86
Moving on to the AMA (Actual Mechanical Advantage). AMA is the ratio of the output force to the input force. In this case, the input force is the force exerted by the student, which is given as 70 N. The output force is the weight of the cart and boxes:
Output force = Weight = Mass × Gravity
Output force = (Total mass) × Gravity
Output force = 30 kg × 9.8 m/s^2
Output force = 294 N
Therefore, the AMA is:
AMA = Output force / Input force
AMA = 294 N / 70 N
AMA ≈ 4.2
Next, let's calculate the efficiency. Efficiency in this context is the ratio of work out to work in, multiplied by 100% to get the percentage. Therefore, the efficiency is given by:
Efficiency = (Work out / Work in) × 100%
Efficiency = (529.2 J / 126 J) × 100%
Efficiency ≈ 420%
Finally, let's determine the power. Power is the rate at which work is done or work done per unit time. Therefore, power can be calculated as:
Power = Work / Time
Power = (Work out / Time)
Power = (529.2 J / 42 s)
Power ≈ 12.60 W
To summarize:
- Work in = 126 J
- Work out = 529.2 J
- IMA ≈ 3.86
- AMA ≈ 4.2
- Efficiency ≈ 420%
- Power ≈ 12.60 W