Question

A particle slides freely on a flat surface, attached to a massless string that passes through a hole in the surface. At a given moment, the particle is moving in a circle of radius $r_0$ with angular velocity ${\omega }_0$. Assume that the string is pulled through the hole at a constant rate. Find an expression for the tension in the string as a function of the length $r$ of string left on the surface.

Short answer

Tension is given by $T=m\frac{r_0^4}{r^3}{\omega }_0^2$.

Step-by-step solution

Understanding Circular Motion and Angular Momentum

Given that the particle moves in a circle, its motion can be described using angular momentum. Initially, the particle has an angular momentum given by $L=m{r}_0^2{\omega }_0$, where $m$ is the mass of the particle, $r_0$ is the initial radius, and ${\omega }_0$ is the initial angular velocity. Because angular momentum is conserved in the absence of external torques, $L=m{r}^2\omega$ must remain constant as the string is pulled.

Deriving Angular Velocity as a Function of r

Using the conservation of angular momentum, we have $m{r}^2\omega =m{r}_0^2{\omega }_0$. Solving for $\omega$, the angular velocity as the radius decreases to $r$, gives us $\omega =\frac{r_0^2}{r^2}{\omega }_0$. This describes how the angular velocity changes as the string is pulled and the radius decreases.

Relating Centripetal Force to Tension in the String

The tension $T$ in the string provides the centripetal force required to keep the particle in circular motion. The centripetal force $F_c$ is given by $F_c=mr{\omega }^2$. Since tension is the force acting on the particle, $T=F_c$.

Substitute for Angular Velocity in Centripetal Force

Substitute the expression for $\omega$ from Step 2 into the expression for the centripetal force. This gives: $T=mr{\left(\frac{r_0^2}{r^2}{\omega }_0\right)}^2=mr\frac{r_0^4}{r^4}{\omega }_0^2=m\frac{r_0^4}{r^3}{\omega }_0^2$. This is the expression for tension as a function of the radius $r$.

Key concepts

Angular Momentum Conservation

Angular momentum is a key concept in understanding circular motion, especially in systems like the one described in the exercise with a particle attached to a string. Angular momentum, denoted as $L$, is calculated using the formula $L=m{r}^2\omega$, where $m$ is the mass of the particle, $r$ is the radius, and $\omega$ is the angular velocity.
This principle states that in the absence of external torques, the angular momentum of a system remains constant. This is a convenient assumption here as it simplifies calculations.

• Conserved Quantity: It doesn’t change with time unless influenced by external factors.
• Initial Angular Momentum: At the start, it is $L=m{r}_0^2{\omega }_0$.
• Constant During Motion: Even as the string is pulled, it must satisfy $m{r}^2\omega =m{r}_0^2{\omega }_0$.

Understanding this is vital as it allows us to relate changes in radius to changes in angular velocity.

Angular Velocity

Angular velocity describes how fast an object rotates around a fixed axis. It's represented by $\omega$, and in the context of this particle, it's key to understanding how its speed changes as the radius changes.
From the conservation of angular momentum, we derive that the angular velocity changes according to the formula: $$\omega =\frac{r_0^2}{r^2}{\omega }_0$$ This shows:

• Inverse Relationship: As the radius $r$ decreases, the angular velocity $\omega$ increases.
• Initial Condition: At $r=r_0$, $\omega ={\omega }_0$.

This change occurs because the total angular momentum must remain constant, governed by the shape and size of the system changing but not its momentum.

Centripetal Force

Centripetal force is the necessary force to keep an object moving in a circular path. For the particle in our problem, this force is provided by the tension in the string.
The formula for centripetal force $F_c$ in terms of the particle is given by:$$F_c=mr{\omega }^2$$ As the tension provides this force, they are equal. Key points include:

• Direction: The force always points towards the center of the circle.
• Dependence on Radius and Velocity: As the radius decreases, the tension or centripetal force changes according to $\omega$.

This force is what keeps the particle on its circular path despite the shrinking radius.

Tension in String

The tension in the string is the force acting on the particle that maintains its circular motion. It is essentially the centripetal force required to keep the particle moving in the circle.By substituting the angular velocity expression into the centripetal force equation, we derive the tension in the string as:$$T=m\frac{r_0^4}{r^3}{\omega }_0^2$$ This expression shows:

• Variation with Radius: As the radius $r$ decreases, the tension increases significantly, demonstrating an inverse cube relationship.
• Dependence on Initial Conditions: The expression also depends on the initial radius $r_0$ and the initial angular velocity ${\omega }_0$.

This mathematical result highlights how the system's dynamics change due to the tension adjusting as the radius becomes smaller.