The curve [tex]y = sinx\sqrt{3cosx},
0\leq x\leq \pi /2[/tex] rotates around the x-axis and creates a homogenous rotational body K.
a) Decide the volume of K.
b) Decide the center of mass for K.
(The x-coordinate of the center of mass is [tex]X_{T} = \frac{1}{m}\int_{K}^{} x dm[/tex], where m is the mass of of K.)
Pir said:Homework Statement
Homework Equations
[tex]X_{T} = \frac{1}{m}\int_{K}^{} x dm[/tex]
The Attempt at a Solution
I have decided a) and I got the volume to be ∏ volume units. I need help with b).
I try to use the formula and I get this:
[tex]X_{T} = \frac{1}{\pi}\int_{K}^{} 3\pi x sin^2xcosx dx[/tex]
But I need help how to solve this integral (if it's correct?). Please help me with this, how do I solve this integral?
Pir said:Is that formula correct to begin with?
I tried integrating by parts but I couldn't solve it. Please help by writing how to do it. I have a test in two days and I need to know how to solve this before then and it takes forever if I have to ask for one step at a time when there might be several questions.
Dick said:Yes, it looks correct. Try parts using u=x dv=sin(x)^2*cos(x)dx.
Pir said:I don't understand, please write how to do it. You mean substitute x with u? That doesn't really change anything, there are still three factors.
Pir said:What happens with the pi?
And why isn't the x integrated with the rest? Shouldn't it be 1/2 x^2?
Pir said:Yes the pi can be factored out but DID YOU NOTICE THAT THERE'S ALSO A 1/PI, DICK? What happens with that?
The center of mass is a point in a system or object where the mass is evenly distributed in all directions. It is also known as the center of gravity.
Determining the center of mass allows us to understand how an object will move and behave in response to external forces. It is also crucial in designing stable structures and machines.
The center of mass can be calculated by finding the weighted average of the individual masses and their respective distances from a reference point. This is often done using mathematical equations or with physical experiments.
The location of the center of mass is affected by the shape, density, and distribution of mass within an object. External forces such as gravity and friction can also influence the position of the center of mass.
Determining the center of mass has many practical applications, including designing stable buildings and bridges, understanding the movement of objects in space, and improving the performance of vehicles such as airplanes and cars. It is also important in sports, such as balancing in gymnastics or throwing a discus.