Question

Two bodies of equal masses revolve in circular orbits of radii ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ with the same period Their centripetal forces are in the ratio. (A) ${\left({\mathrm{R}}_2/{\mathrm{R}}_1\right)}^2$ (B) $\left({\mathrm{r}}_1/{\mathrm{r}}_2\right)$ (C) ${\left({\mathrm{R}}_1/{\mathrm{R}}_2\right)}^2$ (D) ${\sqrt{(}}_1{R}_2)$

Short answer

The ratio of the centripetal forces is $\frac{R_1}{R_2}$, which is the answer (B).

Step-by-step solution

Recall the centripetal force formula

The formula for centripetal force is given by: $F=m\frac{v^2}{r}$ Where $F$ is the centripetal force, $m$ is the mass of the body, $v$ is the tangential velocity, and $r$ is the radius of the circular orbit.

Express velocity in terms of period

Given that the two bodies have the same period ($T$), we can express their velocities in terms of the period. The relation between tangential velocity, radius, and period is given by: $v=\frac{2\pi r}{T}$

Substitute velocity in the centripetal force formula

We can now substitute the expression of tangential velocity from Step 2 into the centripetal force formula from Step 1 for both orbits. Body 1: $F_1=m{\left(\frac{2\pi R_1}{T}\right)}^2\cdot \frac{1}{R_1}$ Body 2: $F_2=m{\left(\frac{2\pi R_2}{T}\right)}^2\cdot \frac{1}{R_2}$

Simplify and find the ratio of the centripetal forces

We can simplify each equation and then find the ratio of the centripetal forces. $F_1=\frac{4{\pi }^{2}m{R}_1}{T^2}$ $F_2=\frac{4{\pi }^{2}m{R}_2}{T^2}$ Now find the ratio $\frac{F_1}{F_2}$: $\frac{F_1}{F_2}=\frac{\frac{4{\pi }^{2}m{R}_1}{T^2}}{\frac{4{\pi }^{2}m{R}_2}{T^2}}$ Cancel out the common terms: $\frac{F_1}{F_2}=\frac{R_1}{R_2}$ So, the ratio of the centripetal forces is $\frac{R_1}{R_2}$, which is the answer (B).

Key concepts

Circular Motion

Circular motion occurs when an object moves in a path defined by a circle. Imagine a car driving around a circular track or the Earth orbiting the sun. In this kind of motion, the speed is constant, but the direction continuously changes. That constant change in direction is what keeps the object in its circular path.

There are a few points to remember about circular motion:

• The object is always accelerating, even if moving at a constant speed, because its direction changes.
• A force is required to keep the object moving in its circular path, which leads us to centripetal force.
• The radius of the circle can affect the properties of the motion, like speed and acceleration.

Understanding these points is essential since all these aspects connect to force, velocity, and time.

Tangential Velocity

Tangential velocity is a concept that describes how fast an object travels along the circular path. It is tangent to the circle at any point, meaning it is at a right angle to the radius at the location of the object.

Let's break it down:

• If you imagine a ball tied to a string and spun in a circle, the tangential velocity would be the speed at which the ball would travel if the string suddenly broke, moving off in a straight line.
• The formula to calculate tangential velocity is given by: $v=\frac{2\pi r}{T}$ where $v$ represents the tangential velocity, $r$ is the radius, and $T$ is the period.
• The larger the radius, the greater the tangential velocity, given the same period.

This velocity intersects with other important concepts, such as how fast something needs to move to stay in a circular path.

Period of Revolution

The period of revolution is the time it takes for an object to make one complete circle around its path. It is often denoted by $T$. Think of it as one full lap, like that of a ferris wheel making one complete round.

A few key points include:

• A shorter period means the object is moving faster. While a longer period indicates slower circular motion.
• It is inversely related to the tangential velocity; as velocity increases for a constant radius, the period decreases.
• When all other factors are constant, the period can help determine other variables like speed and centripetal force applied.

It's fundamental to relate the period to velocity to understand the dynamics of circular motion better.

Centripetal Force Formula

The centripetal force formula is at the core of understanding circular motion. This force is what keeps an object in its circular path, directed towards the center of the circle.

Here's the breakdown:

• Expressed by $F=m\frac{v^2}{r}$, where $F$ is the centripetal force, $m$ is the mass, $v$ is the tangential velocity, and $r$ is the radius.
• This formula shows that the force is proportional to the square of the speed of the object; as speed increases, the force increases significantly.
• Also, as the radius of the circle increases, the force needed decreases, assuming velocity and mass remain constant.
• It's crucial to know that without this force, an object would simply continue in a straight line due to inertia.

Understanding this formula helps grasp why objects continue in circular paths and how changing various factors impacts the required force.