Two particles A and B are in uniform circular motion about a common center. The acceleration of particle A is 4.7 times that of particle B. Particle B takes 2.4 times as long for a rotation as particle A. Th ratio of the radius of the motion of particle A is that of particle B is closest to?

Let's denote the acceleration of particle A as \(a_A\) and the acceleration of particle B as \(a_B\).

Given that the acceleration of particle A is 4.7 times that of particle B, we have:
\[a_A = 4.7a_B \qquad (1)\]

Also, it is given that particle B takes 2.4 times as long for a rotation as particle A, which means the periods of their motions satisfy the relationship:
\[T_B = 2.4T_A \qquad (2)\]

Now, let's consider the relationship between acceleration and radius of circular motion. The centripetal acceleration for a particle in uniform circular motion is given by:
\[a = \frac{v^2}{r}\]

Where:
- \(v\) is the linear velocity of the particle, and
- \(r\) is the radius of the circular motion.

Since the linear velocities of particles A and B will be different in general, we need to consider their ratios of accelerations. Using the above equation, we can rewrite the ratios of accelerations as ratios of linear velocities:
\[\frac{v_A^2}{r_A} = 4.7 \times \frac{v_B^2}{r_B}\]

We can rearrange this equation to isolate the ratio of radii:
\[\frac{r_A}{r_B} = \frac{v_A^2}{4.7v_B^2} \qquad (3)\]

Next, let's consider the periods of motion. The linear velocity can be related to the period by the equation:
\[v = \frac{2\pi r}{T}\]

Using this relation, we can rewrite equation (3) in terms of the periods:
\[\frac{r_A}{r_B} = \frac{\left(\frac{2\pi r_A}{T_A}\right)^2}{4.7 \times \left(\frac{2\pi r_B}{T_B}\right)^2}\]

Simplifying this equation, we obtain:
\[\frac{r_A}{r_B} = \frac{T_B^2}{4.7T_A^2} \qquad (4)\]

Substituting equation (2) into equation (4), we get:
\[\frac{r_A}{r_B} = \frac{(2.4T_A)^2}{4.7T_A^2}\]

Simplifying further:
\[\frac{r_A}{r_B} = \frac{5.76T_A^2}{4.7T_A^2}\]

The \(T_A^2\) terms cancel out:
\[\frac{r_A}{r_B} = \frac{5.76}{4.7}\]

Evaluating this expression, we find:
\[\frac{r_A}{r_B} \approx 1.226\]

Therefore, the ratio of the radius of the motion of particle A to that of particle B is approximately 1.226.

To solve this problem, we need to relate the given information about the acceleration and time taken for rotation to find the ratio of the radii of the circular motion of particles A and B.

Let's denote the acceleration of particle A as "aA" and the acceleration of particle B as "aB". We are given that the acceleration of particle A is 4.7 times that of particle B, so we can write:

aA = 4.7 * aB (Equation 1)

We are also given that particle B takes 2.4 times as long for a rotation as particle A. This means the time taken for one rotation by particle B (TB) is 2.4 times the time taken for one rotation by particle A (TA):

TB = 2.4 * TA (Equation 2)

In uniform circular motion, the centripetal acceleration is given by:

a = (v^2) / r

where "v" is the velocity of the particle and "r" is the radius of the circular motion.

From Equation 1, we know that aA = 4.7 * aB. Therefore, the ratio of the velocities (vA/vB) will also be 4.7:

vA / vB = 4.7 (Equation 3)

We can rewrite the centripetal acceleration equation as:

a = (v^2) / r
v^2 = ar
v^2 = (aA * rA) = (aB * rB)

Using Equation 3, we can substitute vA = 4.7 * vB into the equation:

(4.7 * vB)^2 = (aB * rB)

From Equation 2, we know that TB = 2.4 * TA.

Since the velocity (v) of a particle in uniform circular motion is given by:

v = 2πr / T

where "T" is the time period for one rotation, we can write:

vA = 2πrA / TA (Equation 4)
vB = 2πrB / TB (Equation 5)

Substituting Equations 4 and 5 into Equation 3, we get:

(2πrA / TA) / (2πrB / TB) = 4.7

Cancelling out the common terms, we have:

(TB / TA) * (rA / rB) = 4.7

Using Equation 2, TB = 2.4 * TA, we can substitute this into the equation:

(2.4 * TA / TA) * (rA / rB) = 4.7

Simplifying, we find:

2.4 * (rA / rB) = 4.7

Dividing both sides by 2.4, we get:

(rA / rB) = 4.7 / 2.4

So, the ratio of the radius of the motion of particle A to that of particle B is closest to 1.958.

Therefore, the answer is approximately 1.958.

0.87