The scale of a spring balance reading from 0 to 250N has a length of 10.0 cm. A fish hanging from the bottom of the spring oscillates vertically at a frequency of 2.85 Hz.
Ignoring the mass of the spring, what is the mass m of the fish?
(and why the heck do we have a fish on a spring?)
I am uncertain when is the length 10cm? When stretched with a 250N force?
if so, k=250/.1=2500N/m
If you are in a calculus based physics course, this is of some interest
http://www.kettering.edu/~drussell/Demos/SHO/mass.html
If you are in a plug and chug physics course, then
f=1/2PI *sqrt(k/m) and you solve for m.
thanks! i used the two equations for w0 and set them equal to each other and solved for m and it worked!
thanks!
To find the mass of the fish, we need to use the formula for the frequency of an oscillating spring:
f = 1 / (2 * π) * √(k / m)
Where:
f represents the frequency
k represents the spring constant
m represents the mass of the fish
First, let's convert the frequency given in Hz to angular frequency in radians per second (ω):
ω = 2 * π * f
= 2 * π * 2.85
≈ 17.94 rad/s
Since the length of the scale on the spring balance is 10.0 cm, we can calculate the spring constant (k) using Hooke's Law:
k = (m * g) / x
Where:
g represents acceleration due to gravity (approximated as 9.8 m/s^2)
x represents the displacement from the equilibrium position (given as 10.0 cm or 0.1 m)
We can rearrange the formula to solve for m:
m = (k * x) / g
Now we can substitute the value of x (0.1 m) into the equation:
m = (k * 0.1) / g
Since we have a spring balance reading from 0 to 250N, which corresponds to a displacement from the equilibrium position of 0.1 m, we can determine the spring constant (k) as follows:
k = F / x
= 250 / 0.1
= 2500 N/m
Substituting this value into the equation:
m = (2500 * 0.1) / 9.8
= 2.55 kg
Therefore, the mass of the fish hanging from the bottom of the spring is approximately 2.55 kg.
As for why there is a fish on a spring, it's likely a hypothetical scenario designed to help understand the principles of oscillation and the behavior of springs.
To find the mass of the fish, we can use the concept of simple harmonic motion. The frequency of oscillation, in this case, can be related to the spring constant and the mass of the fish.
First, let's convert the frequency to angular frequency (ω). The angular frequency can be found using the formula:
ω = 2πf
where f is the frequency. Substituting the values, we find:
ω = 2π * 2.85 Hz
ω = 17.91 rad/s
Next, in simple harmonic motion, the angular frequency ω is related to the spring constant k and the mass m by the formula:
ω = √(k/m)
Since we are ignoring the mass of the spring, we can say that the spring constant k is proportional to the length of the scale on the spring balance. In this case, the scale has a length of 10.0 cm. We need to convert the length to meters by dividing by 100:
Length = 10.0 cm = 0.1 m
Now, we can substitute the values and solve for m:
ω = √(k/m)
17.91 rad/s = √(k/m)
Square both sides:
319.7681 rad^2/s^2 = k/m
Now, we have everything we need to find the mass m. We know that the scale on the spring balance ranges from 0 to 250 N. The force F acting on the fish can be related to the spring constant k and the displacement x of the spring by the formula:
F = kx
At the maximum displacement, the force F is equal to the weight of the fish, which is mg, where g is the acceleration due to gravity. So we have:
F = mg
Substituting the formula for force, we have:
mg = kx
Since the maximum force occurs at the maximum displacement, we can say that the weight of the fish is equal to k times the maximum displacement. The maximum displacement can be calculated using the length of the scale:
Maximum Displacement = Length/2 = 0.1 m / 2 = 0.05 m
Now we can calculate the spring constant k:
k = F/x = 250 N / 0.05 m = 5000 N/m
Finally, we can plug in the values for k and solve for m:
319.7681 rad^2/s^2 = k/m
319.7681 rad^2/s^2 = (5000 N/m) / m
Divide both sides by 5000 N/m:
0.06395362 m = 1/m
Take the reciprocal of both sides:
m = 1 / 0.06395362 m
m = 15.642 kg
Therefore, the mass m of the fish is approximately 15.642 kg.
As for why we have a fish on a spring in this example, it's simply a hypothetical situation used to demonstrate the application of physics principles. It helps to illustrate how simple harmonic motion, spring constants, and oscillation frequencies can be related to determine the mass of an object.