The resilin in the body of a flea has a force constant of about 28 kg, and when the flea its jumping legs, the resilin associated with each leg is stretched by approximately 0.10 mm.
Part A
Given that the flea has a mass of 0.56 mg, and that two legs are used in a jump, estimate the maximum height a flea can attain by using the energy stored in the resilin. (Assume the resilin to be an ideal spring.)
Well, let's crunch some numbers, or should I say, jump into them!
First, we need to calculate the potential energy stored in the resilin. This can be done using the formula:
PE = (1/2)kx²
Where:
PE is the potential energy
k is the force constant
x is the displacement
Now, we know that the force constant (k) is 28 kg, and the displacement (x) is 0.10 mm.
This means the potential energy stored in each leg is:
PE = (1/2)(28 kg)(0.10 mm)²
But remember, the flea has two legs, so we need to double that:
PE_total = 2 * (1/2)(28 kg)(0.10 mm)²
Got your calculator handy? Okay, here we go:
PE_total = (2)(0.5)(28 kg)(0.10 mm)²
Feel free to calculate that and then we'll continue with the fun!
To estimate the maximum height a flea can attain using the energy stored in the resilin, we can use the principle of conservation of energy.
Step 1: Calculate the potential energy stored in the resilin.
The potential energy stored in a spring is given by the equation:
Potential Energy (PE) = (1/2) * k * x^2
where k is the force constant and x is the displacement of the spring.
Given:
Force constant (k) = 28 kg
Displacement (x) = 0.10 mm = 0.10 * 10^(-3) m
PE = (1/2) * 28 * (0.10 * 10^(-3))^2
Step 2: Calculate the total potential energy stored in the two legs.
Since each leg has resilin associated with it, and two legs are used in a jump, the total potential energy stored in the two legs can be calculated as:
Total PE = 2 * PE
Substituting the calculated value of PE into the equation:
Total PE = 2 * [(1/2) * 28 * (0.10 * 10^(-3))^2]
Step 3: Calculate the maximum height.
Since potential energy is converted to gravitational potential energy when the flea jumps, the maximum height it can attain is given by:
Maximum Height (h) = Total PE / (mass * gravitational acceleration)
Given:
Mass of flea = 0.56 mg = 0.56 * 10^(-6) kg
Gravitational acceleration (g) = 9.8 m/s^2
Substituting the values into the equation:
Maximum Height (h) = [2 * (1/2) * 28 * (0.10 * 10^(-3))^2] / (0.56 * 10^(-6) * 9.8)
Now, you can calculate the value by simplifying the expression.
To estimate the maximum height a flea can attain using the energy stored in the resilin, we need to use the principle of conservation of energy.
The potential energy stored in the resilin can be calculated using the formula:
Potential Energy = 1/2 * force constant * (stretch)^2
Given:
Force constant (k) = 28 kg
Stretch (Δx) = 0.10 mm (which we need to convert to meters)
First, let's convert the stretch from millimeters to meters:
0.10 mm = 0.10 * 10^(-3) m = 0.0001 m
Now, let's calculate the potential energy stored in one leg of the flea:
Potential Energy = 1/2 * k * (Δx)^2
= 1/2 * 28 * 0.0001^2
= 0.00014 J (Joules)
Since two legs are used in a jump, the total potential energy stored in the resilin is twice this value:
Total Potential Energy = 2 * 0.00014 J
= 0.00028 J
Next, we can use the principle of conservation of energy to relate the potential energy stored in the resilin to the maximum height the flea can attain.
At the maximum height, all the potential energy will be converted into gravitational potential energy. So, we can equate the two:
Potential Energy = gravitational potential energy
0.00028 J = m * g * h
Where:
m = mass of the flea (0.56 mg = 0.56 * 10^(-6) kg)
g = acceleration due to gravity (9.8 m/s^2)
h = maximum height
Rearranging the equation to solve for h:
h = Potential Energy / (m * g)
= 0.00028 J / (0.56 * 10^(-6) kg * 9.8 m/s^2)
≈ 0.000057 m
Therefore, the estimated maximum height the flea can attain using the energy stored in the resilin is approximately 0.000057 meters, or 0.057 mm.