Linear Mass Density: Find 8.9866e-4 m/s to g/m

[brianna0404]

Linear Mass Density!

Homework Statement

Find linear mass density using a graph ln(v) vs ln(T), from Velocity=squareroot Tension/linear density, take natural log ln(v) = 1/2 ln(T)-1/2ln(m). our professor told us to find it this way cause 1/2ln(m) is the y-intercept of the best fit line of five trials

Homework Equations

so y-intercept=-1/2ln(m). for y intercept I got 3.5073 from an experiment done in lab.

The Attempt at a Solution

I got 8.9866 * 10^-4 for linear mass density. but after all that my main question is what are the units? would they be m/s? if so how do I convert it to g/m?

Thanks!

 

[EkaterinaAvd]

it depends on units in which v and T are given. you have to extract units from formula Velocity=squareroot Tension/linear density.

Units of right and left parts of equation must be the same.

 

[brianna0404]

the units for tension are Newtons and the units for velocity are hertz times meters, which is just m/s. So i didn't know if taking the natural log of tension and velocity changes the units or are they the same?

 

[EkaterinaAvd]

logarithm equation and equation without logarithm are eqivalent
if [v]=m/s, it doesn't mean that [ln(v)]=m/s of course. They are two different numbers representing the same velocity and can't have the same dimension.
but if you are consistent with formulas you have in your calculations, you will get right units from the initial equation.

because

ln(v) = 1/2 ln(T)-1/2ln(m)

is the same as

ln(v)=ln(sqrt(T/m))

is the same as

v=sqrt(T/m), m=T/v*v.

So, if [T]=N, [v]=m/s, you can easily extract [m]. And it is not m/s.

 

[asteriatic]

I would first commend you on your thorough approach to finding the linear mass density. It is important to use data and mathematical relationships to determine accurate values in scientific experiments.

To answer your question about the units, the units for linear mass density are typically expressed as mass per unit length, such as grams per meter (g/m) or kilograms per meter (kg/m). In your case, you have calculated a value of 8.9866 * 10^-4 m/s for the linear mass density. However, this unit is not consistent with the standard units for linear mass density.

To convert from m/s to g/m, you will need to use the relationship between mass and velocity. The equation for linear mass density, as given by your professor, is v = √(T/λ), where v is velocity, T is tension, and λ is linear mass density. This can also be written as λ = T/v^2. In this case, velocity is given in m/s, so you will need to convert it to m^2/s^2 in order to cancel out the units of velocity.

To convert from m/s to m^2/s^2, you can simply multiply by m, since m/s is the same as m/s^2. This gives you a value of 8.9866 * 10^-4 m^2/s^2 for the linear mass density. To convert this to g/m, you will need to multiply by a conversion factor of 1 g/1000 m^2. This gives you a final value of 8.9866 * 10^-7 g/m for the linear mass density.

In summary, the units for linear mass density are typically expressed as mass per unit length, such as g/m or kg/m. To convert from m/s to g/m, you will need to use the relationship between mass, velocity, and length, and apply appropriate conversion factors. I hope this helps clarify the units for you.