Find Mass & Center of Mass for a Rod of Length 38.5 cm

[chocolatelover]

Homework Statement

A rod of length 38.5 cm has linear density (mass-per-length) given by the following equation, where x is the distance from one end.

λ=50.0g/m+16.5g/m^2

a. What is its mass in g?

b. How far from the x=0 end is its center of mass in m?

Homework Equations

The Attempt at a Solution

For part a., could I just divide by the mass?

Could someone please show me how to do this problem? I really don't understand what it's asking for.

Thank you very much

 

[CrazyIvan]

For part A:

By definition,
$$mass = \lambda \cdot length$$

Since linear density isn't constant, you will need to integrate. But first you need an expression for how the mass of an infinitesimal length of the rod. Can you figure out what this is?

After you have an expression for how $$dm$$ relates to $$dx$$, you can integrate both sides to get the total mass.For part B, start with the general expression for center of mass.

 

[Moetasim]

.I can provide a response to your question. To find the mass of the rod, we need to integrate the linear density function over the entire length of the rod. This can be done by using the formula:

m = ∫λdx

where m is the mass, λ is the linear density function, and x is the distance from one end of the rod.

In this case, the linear density function is given by:

λ=50.0g/m+16.5g/m^2

So, the mass of the rod can be calculated as:

m = ∫(50.0g/m+16.5g/m^2)dx

= 50.0x + 16.5x^2/2 + C

where C is the constant of integration.

Now, to find the center of mass, we can use the formula:

x_cm = ∫xλdx/∫λdx

where x_cm is the distance from the x=0 end to the center of mass.

So, the center of mass of the rod can be calculated as:

x_cm = ∫x(50.0g/m+16.5g/m^2)dx/∫(50.0g/m+16.5g/m^2)dx

= (50.0x^2/2 + 16.5x^3/6 + C)/ (50.0x + 16.5x^2/2 + C)

= (50.0x^2 + 16.5x^3)/ (100.0x + 16.5x^2 + C)

To find the value of C, we can use the known length of the rod (38.5 cm) and the fact that the center of mass should be at the midpoint of the rod (x_cm = 38.5/2 = 19.25 cm). So, we can write the following equation:

19.25 = (50.0x^2 + 16.5x^3)/ (100.0x + 16.5x^2 + C)

Solving this equation for C, we get:

C = 9.2x^2 - 19.25x + 19.25

Now, we can substitute this value of C in our previous equation for x_cm and solve for x_cm:

x_cm