A uniform thin wire is bent into a semicircle of radius r. Determine the coordinates of its center of mass with respect to an origin of coordinates at the center of the "full" circle.
Because wire is uniform x (sub cm) = 0
mass=dm
length=dl
coordinates (x,y) = (r cox theta, r sin theta)
To determine the coordinates of the center of mass of the semicircle, you need to find the average position of all the infinitesimally small masses that make up the wire.
Let's break down the process step by step:
1. Define the coordinate system: Since we want to find the coordinates relative to the center of the "full" circle, we can set the origin at the center of the circle and use polar coordinates with the radial distance r and angle theta.
2. Express the infinitesimally small mass element: The wire is uniform, so the mass per unit length, dm, is constant. We can express the infinitesimally small length element, dl, as r dθ, where dθ is the differential angle along the semicircle.
3. Calculate the position vector: The position vector of an infinitesimally small mass element can be expressed as (x, y) = (r cos(theta), r sin(theta)) in polar coordinates.
4. Calculate the total mass: Integrate dm along the wire to obtain the total mass, M. Since dm is constant, M = dm * L, where L is the total length of the wire.
5. Calculate the coordinates of the center of mass: The x-coordinate of the center of mass, x_cm, is given by the formula:
x_cm = (1/M) * ∫ (x * dm)
Substituting the expression for x in terms of theta, we have:
x_cm = (1/M) * ∫ (r cos(theta) * dm)
Similarly, the y-coordinate of the center of mass, y_cm, is given by:
y_cm = (1/M) * ∫ (y * dm) = (1/M) * ∫ (r sin(theta) * dm)
6. Perform the integrations: Substitute dm = (M/L) * r dθ into the x_cm and y_cm formulas. Then integrate from θ = 0 to θ = π, which covers the semicircle.
x_cm = (1/M) * ∫ (r cos(theta) * (M/L) * r dθ)
y_cm = (1/M) * ∫ (r sin(theta) * (M/L) * r dθ)
Simplifying these expressions, we have:
x_cm = (1/L) * ∫ (r^2 cos(theta) * dθ)
y_cm = (1/L) * ∫ (r^2 sin(theta) * dθ)
7. Evaluate the integrals: Integrate each term using the appropriate trigonometric identities and limits of integration θ = 0 to θ = π.
x_cm = (1/L) * ∫ (r^2 cos(theta) * dθ) = (r/L) * ∫ (r cos^2(theta) * dθ)
y_cm = (1/L) * ∫ (r^2 sin(theta) * dθ) = (r/L) * ∫ (r sin^2(theta) * dθ)
Use trigonometric identities for cos^2(theta) and sin^2(theta) to simplify the integrals further.
8. Solve the integrals: Evaluate the integrals and simplify the expressions obtained for x_cm and y_cm.
After performing the integration, you should obtain the final formulas for x_cm and y_cm in terms of r and L.
Calculating the center of mass of the wire involves some calculus and trigonometry. However, by following these steps, you can arrive at the coordinates of the center of mass relative to the origin at the center of the "full" circle.