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
I will be happy to critique your thinking.
To find the required values, let's break down the problem step by step:
1. Work In:
Work done on an object can be calculated using the equation W = F × d × cos(θ), where W is the work done, F is the force applied, d is the distance covered, and θ is the angle between the force and the direction of motion.
In this case, the force applied is 70 N, and the distance covered is the length of the ramp. To find the distance, we can use trigonometry. The length of the ramp can be calculated using the formula L = h / sin(θ), where L is the length of the ramp, h is the height of the platform, and θ is the angle of inclination.
Given that the platform's height (h) is 1.8 m and the angle (θ) is 15 degrees, we can calculate the length (L) of the ramp:
L = h / sin(θ) = 1.8 / sin(15) = 6.89 m
Now we can calculate the work done (W) using the formula W = F × d × cos(θ):
W = 70 × 6.89 × cos(15) = 442.4 J
Therefore, the work done on the cart is 442.4 Joules (J).
2. Work Out:
The work done by the gravitational force on the cart is equal to the change in potential energy. Since the cart is raised to a height of 1.8 m, we can calculate the work done by the gravitational force using the formula W = m × g × h, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The total mass of the cart and two boxes of books is 6 kg (cart) + 2 (12 kg) = 30 kg.
Now we can calculate the work done by the gravitational force:
W = m × g × h = 30 × 9.8 × 1.8 = 529.2 J
Therefore, the work done by the gravitational force on the cart is 529.2 Joules (J).
3. IMA (Ideal Mechanical Advantage):
The ideal mechanical advantage (IMA) of a machine is the ratio of the output force to the input force in the absence of friction. In this case, the ramp acts as a simple machine.
The IMA of an inclined plane (ramp) can be calculated using the formula IMA = L / h, where L is the length of the ramp and h is the height of the ramp.
Using the values we found earlier, the IMA is calculated as follows:
IMA = L / h = 6.89 / 1.8 = 3.828
Therefore, the IMA of the ramp is approximately 3.828.
4. AMA (Actual Mechanical Advantage):
The actual mechanical advantage (AMA) takes into account the effects of friction and is usually less than the IMA. In this case, the AMA can be calculated using the formula AMA = Force out / Force in.
Given that the force applied (Force in) is 70 N and the weight of the cart and books (Force out) is the gravitational force, which is approximately 30 kg × 9.8 m/s^2 = 294 N, we can calculate the AMA as follows:
AMA = 294 N / 70 N = 4.2
Therefore, the AMA of the system is approximately 4.2.
5. Efficiency:
The efficiency of a machine is the ratio of the useful work output to the work input, expressed as a percentage. The formula for efficiency is Efficiency = (Work out / Work in) × 100.
Using the values we calculated earlier:
Efficiency = (529.2 J / 442.4 J) × 100 = 119.6%
Therefore, the efficiency of the system is approximately 119.6%.
6. Power:
Power is the rate at which work is done or energy is transferred. It can be calculated using the formula Power = Work / Time.
Given that the work done is 442.4 J and the time taken is 42 seconds, we can calculate the power as follows:
Power = 442.4 J / 42 s = 10.54 W
Therefore, the power required to move the cart up the ramp is approximately 10.54 Watts (W).