Question

1. 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 i.e where k is a proportionality constant. Determine the values of n and m and write the equation for the acceleration. [5]

Answer 1

Answer on Question # 62711 – Physics – Mechanics | Relativity

1. 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 $r^{\wedge}n$, and some power of $v$, say $v^{\wedge}m$ i.e. where $k$ is a proportionality constant. Determine the values of $n$ and $m$ and write the equation for the acceleration. [5].

Solution:

Let acceleration be as follows:

$$a = k r^n v^m.$$

Since we know the dimensions of acceleration, radius and velocity, we can write a dimensional equation:

$$L / T^2 = L^n (L / T)^m = L^{n + m} / T^m.$$

The dimensional equation is balanced under conditions: $n + m = 1$ and $m = 2$. Therefore, $n = -1$. The equation for acceleration is as follows:

$$a = k r^{-1} v^2 = k \frac{v^2}{r}.$$

Answer: $a^2 = k \frac{v^2}{r}$, $n = -1$, $m = 2$.

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