The distance moved by a particle in time from centre of ring under the influence of its gravity is given by `x=a sin omegat` where a and `omega` are constants. If `omega` is found to depend on the radius of the ring (r), its mass (m) and universal gravitation constant (G), find using dimensional analysis an expression for `omega` in terms of r, m and G.

Answer

To find the expression for \(\omega\) in terms of \(r\), \(m\), and \(G\) using dimensional analysis, we will follow these steps:

### Step 1: Identify the dimensions of the variables
- The dimension of \(\omega\) (angular frequency) is given by:
\[ [\omega] = T^{-1} \]
- The dimension of the radius \(r\) is:
\[ [r] = L \]
- The dimension of mass \(m\) is:
\[ [m] = M \]
- The dimension of the gravitational constant \(G\) is:
\[ [G] = M^{-1}L^3T^{-2} \]

### Step 2: Write the dimensional formula for \(\omega\)
Assuming \(\omega\) depends on \(r\), \(m\), and \(G\), we can express this as:
\[ \omega \propto r^a m^b G^c \]
This leads to:
\[ [\omega] = [r]^a [m]^b [G]^c \]
Substituting the dimensions we identified:
\[ T^{-1} = (L)^a (M)^b (M^{-1}L^3T^{-2})^c \]

### Step 3: Expand the right side
Expanding the right side gives:
\[ T^{-1} = L^a M^b M^{-c} L^{3c} T^{-2c} \]
This simplifies to:
\[ T^{-1} = L^{a + 3c} M^{b - c} T^{-2c} \]

### Step 4: Equate the dimensions
Now we can equate the dimensions on both sides:
1. For \(T\):
\[ -1 = -2c \quad \Rightarrow \quad c = \frac{1}{2} \]
2. For \(L\):
\[ 0 = a + 3c \quad \Rightarrow \quad a + 3 \left(\frac{1}{2}\right) = 0 \quad \Rightarrow \quad a = -\frac{3}{2} \]
3. For \(M\):
\[ 0 = b - c \quad \Rightarrow \quad b - \frac{1}{2} = 0 \quad \Rightarrow \quad b = \frac{1}{2} \]

### Step 5: Substitute back to find \(\omega\)
Now substituting \(a\), \(b\), and \(c\) back into the expression for \(\omega\):
\[ \omega = k r^{-\frac{3}{2}} m^{\frac{1}{2}} G^{\frac{1}{2}} \]
This can be rewritten as:
\[ \omega = k \frac{\sqrt{mG}}{r^{\frac{3}{2}}} \]

### Step 6: Final expression
Thus, the expression for \(\omega\) in terms of \(r\), \(m\), and \(G\) is:
\[ \omega = k \frac{\sqrt{mG}}{r^{\frac{3}{2}}} \]
where \(k\) is a dimensionless constant.