Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say r^n , and some power of v, say v^m the values of n and m and write the simplest form of an equation for the acceleration. a=kr^nv^m

Given that the acceleration \( a \) of a particle moving with uniform speed \( v \) in a circle of radius \( r \) is proportional to some power of \( r \), say \( r^n \), and some power of \( v \), say \( v^m \), we can write the relationship as:<br /><br />\[ a = k r^n v^m \]<br /><br />To determine the values of \( n \) and \( m \), we need to consider the physical context and the units involved.<br /><br />1. **Acceleration (\(a\))**: The unit of acceleration is meters per second squared (\(\text{m/s}^2\)).<br />2. **Radius (\(r\))**: The unit of radius is meters (\(\text{m}\)).<br />3. **Velocity (\(v\))**: The unit of velocity is meters per second (\(\text{m/s}\)).<br /><br />Let's analyze the units:<br /><br />\[ [a] = \text{m/s}^2 \]<br />\[ [r] = \text{m} \]<br />\[ [v] = \text{m/s} \]<br /><br />Substituting these units into the equation \( a = k r^n v^m \):<br /><br />\[ \text{m/s}^2 = k (\text{m})^n (\text{m/s})^m \]<br /><br />This simplifies to:<br /><br />\[ \text{m/s}^2 = k (\text{m}^n) (\text{m}^m \text{s}^{-m}) \]<br />\[ \text{m/s}^2 = k \text{m}^{n+m} \text{s}^{-m} \]<br /><br />For the units to balance, the exponents must add up to the correct powers of \( \text{m} \) and \( \text{s} \):<br /><br />- The exponent of \( \text{m} \) on the right-hand side must be 1 (since the radius is a length and thus has units of meters).<br />- The exponent of \( \text{s} \) on the right-hand side must be -2 (since acceleration is measured in \(\text{m/s}^2\)).<br /><br />Therefore, we have:<br /><br />\[ n + m = 1 \]<br />\[ -m = -2 \]<br /><br />Solving for \( m \):<br /><br />\[ m = 2 \]<br /><br />Substituting \( m = 2 \) into \( n + m = 1 \):<br /><br />\[ n + 2 = 1 \]<br />\[ n = -1 \]<br /><br />Thus, the simplest form of the equation for the acceleration is:<br /><br />\[ a = k r^{-1} v^2 \]<br /><br />This equation indicates that acceleration is inversely proportional to the radius and directly proportional to the square of the velocity.