The above equation demonstrates the relationship between wave speed, tension, and linear mass density. Since the velocity is directly related to the power of the value of tension, if the tension is increased, the speed will increase as well. It also shows that if linear mass density increases, the wave speed decreases because they are indirectly related. One thing to note is that in a standing wave, amplitude and frequency do not play a role in the wave speed.
Design:
The aim of this experiment is to find the linear mass density of a string. We plan on doing this by using the given formula above and using the velocity formula. We can find the velocity of the wave by controlling the wavelength and frequency we use. We can also find the force of tension by using known masses so that the force of tension will be equal to the force of gravity.
Procedure:
Step 1: Set the strong up to the oscillator and hang it over a pulley in which there is a hanging mass. The force of the hanging mass will be considered to be the force of tension because the mass is at its equilibrium position. Step 2: Set the oscillator to 12Hz so that you can see the wave clearly. Measure the length of the string from the pulley and the amount of nodes given by the wave. Step 3: Use the wavelength and frequency to find the velocity. Step 4: Use the force of gravity by the hanging mass as the force of tension and solve for the linear mass density using the velocity and force of tension.
Calculations:
The velocity equations were manipulated into the equation: μ=Ft/(λ*ƒ)^2. Since we had the frequency of 12Hz and a wavelength of 2 meters, we could solve for the velocity. With the velocity and the given force of tension by the mass, you could solve for the linear mass density. Once solved we find linear mass density of the string is 0.00347kg/m
Conclusion:
Based on our results, I am sure that the linear density that we found is extremely close to the real linear density of the string. We only used one force of tension for the experiment but the value we got makes sense for the linear density to be very low. This is because the string's weight is close to negligible and it is very long.
Evaluation:
One area of uncertainty was the frictional force applied by the pulley onto the string. This could have affected our calculated force of friction because we considered all frictional force to be negligible. Another area of uncertainty was our final results for the linear mass density. Since we only used one force of tension for the experiment, we cannot be sure that our results are accurate.
Improving the Investigation:
To find more accurate results, we could have used different masses for different forces of tension. If we had more time, we could have used different harmonics to achieve even more accurate results for the linear mass density value.