Question

The linear mass density of ring is given as $\lambda = \lambda_0 \sin\theta$. Choose the correct statement (s).

• A. The COM of the ring is at $(0, \frac{\pi R}{8})$ m
• B. The COM of the ring is at $(0, \frac{\pi R}{4})$ m
• C. The COM of the ring is at $(0, \frac{\pi R}{2})$ m
• D. The COM of the ring exist on y axis.

Answer 1

Answer: (B), (D)

The COM of the ring is at $(0, \frac{\pi R}{4})$ m and the COM of the ring exist on y axis.

Answer 2

The center of mass (COM) of a continuous object is found by integration. For a semicircular ring of radius R, we consider an infinitesimal mass element $dm$. The linear mass density is given as $\lambda = \lambda_0 \sin\theta$. The infinitesimal length element on the ring is $dl = R \, d\theta$.

1. Mass element ($dm$): $dm = \lambda \, dl = (\lambda_0 \sin\theta) (R \, d\theta) = \lambda_0 R \sin\theta \, d\theta$.

2. Total mass ($M$): The semicircle spans from $\theta = 0$ to $\theta = \pi$. $M = \int_{0}^{\pi} dm = \int_{0}^{\pi} \lambda_0 R \sin\theta \, d\theta$ $M = \lambda_0 R [-\cos\theta]_{0}^{\pi} = \lambda_0 R (-\cos\pi - (-\cos0)) = \lambda_0 R (-(-1) - (-1)) = 2 \lambda_0 R$.

3. Center of Mass coordinates ($X_{COM}, Y_{COM}$): The coordinates of a point on the ring are $x = R \cos\theta$ and $y = R \sin\theta$. $X_{COM} = \frac{1}{M} \int x \, dm = \frac{1}{M} \int_{0}^{\pi} (R \cos\theta) (\lambda_0 R \sin\theta \, d\theta)$ $X_{COM} = \frac{\lambda_0 R^2}{M} \int_{0}^{\pi} \sin\theta \cos\theta \, d\theta$. Since $\int_{0}^{\pi} \sin\theta \cos\theta \, d\theta = [\frac{\sin^2\theta}{2}]_{0}^{\pi} = 0$, we get $X_{COM} = 0$.

   $Y_{COM} = \frac{1}{M} \int y \, dm = \frac{1}{M} \int_{0}^{\pi} (R \sin\theta) (\lambda_0 R \sin\theta \, d\theta)$ $Y_{COM} = \frac{\lambda_0 R^2}{M} \int_{0}^{\pi} \sin^2\theta \, d\theta$. Using $\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$, the integral is: $\int_{0}^{\pi} \sin^2\theta \, d\theta = \int_{0}^{\pi} \frac{1 - \cos(2\theta)}{2} \, d\theta = \frac{1}{2} [\theta - \frac{\sin(2\theta)}{2}]_{0}^{\pi} = \frac{1}{2} (\pi - 0) = \frac{\pi}{2}$. Substituting $M = 2 \lambda_0 R$: $Y_{COM} = \frac{\lambda_0 R^2}{2 \lambda_0 R} \left(\frac{\pi}{2}\right) = \frac{R}{2} \left(\frac{\pi}{2}\right) = \frac{\pi R}{4}$.

   Thus, the COM of the ring is at $(0, \frac{\pi R}{4})$.

4. Evaluate the statements: (A) The COM of the ring is at $(0, \frac{\pi R}{8})$ m. Incorrect. (B) The COM of the ring is at $(0, \frac{\pi R}{4})$ m. Correct, matches our calculation. (C) The COM of the ring is at $(0, \frac{\pi R}{2})$ m. Incorrect. (D) The COM of the ring exist on y axis. Correct, since $X_{COM} = 0$.

Since the question asks for correct statement(s), both (B) and (D) are correct.