Problem 56 - University Physics with Modern Physics

Question

Weightless Ant. Ant. An ant with mass \(m\) is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length \(\mu\) and is under tension \(F .\) Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wave-length \(\lambda\) propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that \(m\) is so small that the presence of the ant has no effect on the propagation of the wave.

Step-by-Step Solution

Answer

The minimum amplitude for the ant to become weightless is \( A = \frac{g \mu \lambda^2}{4\pi^2 F} \).

Step 1: Understanding Wave Propagation on a Rope

The wave equation for a sinusoidal wave traveling along a rope is given by \( y(x,t) = A \sin\left( \frac{2\pi}{\lambda}(x - vt) \right) \), where \( A \) is the amplitude, \( \lambda \) is the wavelength, and \( v \) is the wave speed. The wave speed \( v \) can be determined from the tension \( F \) and mass per unit length \( \mu \) using the formula \( v = \sqrt{\frac{F}{\mu}} \).

Step 2: Determining Wave Speed

Using the formula for wave speed \( v = \sqrt{\frac{F}{\mu}} \), we have the speed at which the wave propagates along the rope.

Step 3: Analyzing Ant Movement and Acceleration

The ant becomes momentarily weightless when the vertical acceleration it experiences equals the gravitational acceleration \( g \). The vertical acceleration of the wave at any point is given by the second derivative of \( y(x,t) \) with respect to time, \( a_y = \frac{\partial^2 y}{\partial t^2} \).

Step 4: Calculating Vertical Acceleration

The vertical acceleration is calculated by taking the second partial derivative of the wave equation with respect to time: \[ a_y = \frac{\partial^2 y}{\partial t^2} = -A \left( \frac{2\pi}{\lambda} \right)^2 v^2 \sin\left( \frac{2\pi}{\lambda}(x - vt) \right) \]. For maximum acceleration, the sine term becomes \( \pm 1 \).

Step 5: Setting the Acceleration Equal to Gravity

To find the minimum amplitude \( A \, \text{min} \) for which the ant becomes weightless, set the maximum vertical acceleration equal to the acceleration due to gravity, \( g \): \[ A \left( \frac{2\pi}{\lambda} \right)^2 v^2 = g \].

Step 6: Solving for Minimum Amplitude

Rearrange the equation to solve for \( A \): \[ A = \frac{g \lambda^2}{(2\pi)^2 v^2} \]. Substituting \( v = \sqrt{\frac{F}{\mu}} \), we obtain \[ A = \frac{g \lambda^2}{4\pi^2 \frac{F}{\mu}} = \frac{g \mu \lambda^2}{4\pi^2 F} \]. This gives the minimum amplitude.

Key Concepts

Wave Equation

The wave equation describes how waves propagate through a medium, such as a rope. In our case, the equation for a sinusoidal wave traveling along a rope is\[ y(x,t) = A \sin\left( \frac{2\pi}{\lambda}(x - vt) \right) \]where:

• \(y(x,t)\) represents the vertical displacement of the rope at position \(x\) and time \(t\)
• \(A\) is the amplitude of the wave, representing the maximum vertical displacement
• \(\lambda\) is the wavelength, the distance over which the wave's shape repeats
• \(v\) is the wave speed

With these components, the wave equation helps explain how waves carry energy along the rope.

Sinusoidal Waves

Sinusoidal waves are a specific type of wave pattern characterized by their smooth, repetitive oscillations. In our scenario, we see a sinusoidal wave that travels along the rope. This type of wave can be described using sine and cosine functions, making it predictable and uniform. Here are key attributes:

• Amplitude (A): Maximum height or depth the wave reaches from its rest position.
• Wavelength (\(\lambda\)): Distance over which the wave's shape repeats, which is consistent in sinusoidal waves.
• Frequency and period: While not explicitly mentioned, the frequency is the number of wave cycles per second, and the period is the inverse of frequency.

Understanding sinusoidal waves is crucial as it dictates how waves displace a medium, in this case, the rope.

Vertical Acceleration

Vertical acceleration is key to understanding how the ant becomes weightless. For the ant to experience weightlessness, the vertical acceleration of the rope must equal the gravitational acceleration \(g\). The acceleration comes from changes in the wave's vertical displacement over time and is calculated using calculus. By taking the second derivative of the wave equation with respect to time, we find:\[ a_y = -A \left( \frac{2\pi}{\lambda} \right)^2 v^2 \sin\left( \frac{2\pi}{\lambda}(x - vt) \right) \]This formula shows that the vertical acceleration depends on the wave's amplitude, speed, and wavelength. At its peak, this acceleration influences the motion of objects like the ant on the rope.

Weightlessness

Weightlessness occurs when an object experiences no net gravitational force. For the ant on the rope, this happens when the vertical acceleration produced by the wave equals the gravitational acceleration \(g\). At this moment, the ant no longer feels the pull of gravity pulling it downwards. To achieve weightlessness, the rope must move with enough amplitude so that the acceleration force cancels out gravity, achieved when:\[ A \left( \frac{2\pi}{\lambda} \right)^2 v^2 = g \]This equation shows the relationship required for weightlessness, specifically detailing how the wave's properties must align to negate gravity's effect on the ant.

Wave Speed

Wave speed defines how quickly a wave travels along a medium like a rope. It is influenced by the tension \(F\) in the rope and its mass per unit length \(\mu\). The speed is given by the formula:\[ v = \sqrt{\frac{F}{\mu}} \]

• Tension (\(F\)): A measure of the force stretching the rope; increasing tension increases wave speed.
• Mass per unit length (\(\mu\)): Heavier ropes (higher \(\mu\)) reduce wave speed due to additional inertia.

This relationship showcases how physical attributes of the rope impact wave behavior, affecting phenomena like the ant's weightlessness through changes in wave speed.

Mass per Unit Length

Mass per unit length \(\mu\) of the rope is a crucial factor in determining wave behavior. It is defined as the rope's mass divided by its length and influences how easily a wave can travel through it. A higher \(\mu\) implies more inertia, making it harder for waves to propagate quickly, as they have more mass to move. In our case, the speed \(v\) and thus the behavior of the wave are impacted by \(\mu\) through the formula:\[ v = \sqrt{\frac{F}{\mu}} \]

• Lighter ropes (smaller \(\mu\)) enable faster wave propagation.
• More massive ropes (larger \(\mu\)) slow down waves, influencing how the wave's force affects objects like the ant.

By understanding \(\mu\), one can predict how waves will behave, vital for solving problems involving wave mechanics.