A boy on a swing is pulled to a position 2.34 m off the ground and released. At the fastest and lowest point of the swing the boy is 0.50 m off the ground and travelling 3 m/s. The percent efficiency of this swing system is: A) 70 % B) 25% C) 50% D) 10%
[(1/2 m v^2) / (m g h)] * 100% ... [v^2 / (2 g h)] * 100%
3^2 / [2 * 9.8 * (2.34 - 0.50)] * 100%
Well, swinging on a swing can be quite the ride! Let's calculate the efficiency of this swing system.
To do that, we need to find the total mechanical energy of the system at the highest point, which is the potential energy, and compare it to the mechanical energy at the lowest point, which is the sum of the potential and kinetic energies.
At the highest point, the boy is 2.34 m off the ground. So his potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
At the lowest point, the boy is 0.50 m off the ground and traveling at 3 m/s. So the total mechanical energy is the sum of the potential energy and the kinetic energy, given by mgh + (1/2)mv^2, where m is the mass, g is the acceleration due to gravity, h is the height, and v is the velocity.
Now, since the swing system is not perfectly efficient, we know that the mechanical energy at the highest point should be greater than that at the lowest point.
Therefore, let's compare the mechanical energies.
Let's suppose the potential energy at the highest point is "P" and the mechanical energy at the lowest point is "L".
If efficiency is defined as the ratio of useful output to total input, then efficiency = (L/P) * 100%.
If the swing was perfectly efficient (which I'm sure would be quite the magic swing!), then L = P, and the efficiency would be 100%.
But in reality, some energy is lost due to various factors like air resistance and friction, reducing the mechanical energy at the lowest point.
So, based on the given information, we can conclude that the mechanical energy at the highest point is greater than the mechanical energy at the lowest point.
Therefore, the efficiency is less than 100%.
Based on the given answer options, the closest one to this conclusion is D) 10%.
But remember, efficiency is a serious business, so it's important to take this answer with a pinch of salt, or maybe a dash of laughter!
To determine the percent efficiency of the swing system, we need to find the difference in potential energy between the highest and lowest points of the swing and compare it to the initial potential energy.
First, let's calculate the potential energy at the highest point (2.34 m off the ground) of the swing. The potential energy (PE) is given by the formula:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Assuming the mass m is constant, the potential energy at the highest point is:
PE_highest = m * g * h_highest
Next, let's calculate the potential energy at the lowest point (0.50 m off the ground) of the swing. The potential energy at the lowest point is:
PE_lowest = m * g * h_lowest
The difference in potential energy between the highest and lowest points is:
ΔPE = PE_highest - PE_lowest
To calculate the efficiency, we need to compare the change in potential energy (ΔPE) to the initial potential energy (PE_highest). The percent efficiency can be found using the formula:
Efficiency = (ΔPE / PE_highest) * 100
Now, let's plug in the given values:
h_highest = 2.34 m
h_lowest = 0.50 m
Calculating the potential energies:
PE_highest = m * 9.8 m/s^2 * 2.34 m
PE_lowest = m * 9.8 m/s^2 * 0.50 m
Calculating the change in potential energy:
ΔPE = PE_highest - PE_lowest
Calculating the efficiency:
Efficiency = (ΔPE / PE_highest) * 100
Now, let's calculate the values to determine the percent efficiency of the swing system.
To find the percent efficiency of the swing system, we need to compare the initial potential energy (when the boy was pulled to a position 2.34 m off the ground) with the final kinetic energy (when the boy is at the lowest point and traveling at 3 m/s, 0.50 m off the ground).
First, let's calculate the initial potential energy. The potential energy of an object near the Earth's surface is given by the equation:
Potential energy = mass × gravity × height
Given that the boy's position was 2.34 m off the ground, we need to know the boy's mass and the acceleration due to gravity (approximated as 9.8 m/s^2). However, the mass of the boy is not provided in the question. Therefore, we can't directly calculate the percent efficiency of the swing system.
Hence, we don't have enough information to determine the answer.