Question

A mass \(m\) is attached to the end of a rope of length \(r=3\) meters. The rope can only be whirled around at speeds of \(1,10,20\), or 40 meters per second. The rope can withstand a maximum tension of \(T=60\) Newtons. Write a program where the user enters the value of the mass \(m\), and the program determines the greatest speed at which it can be whirled without breaking the rope. Hint: \(T=m v^{2} / r\).

Short answer

Calculate \( v = \sqrt{\frac{60 \times 3}{m}} \) and return the greatest speed from 1, 10, 20, or 40 m/s that is less than or equal to this value.

Step-by-step solution

Understand the Tension Formula

We are given that the tension in the rope is represented by the formula \( T = \frac{m v^{2}}{r} \). Here, \( T \) is the tension, \( m \) is the mass, \( v \) is the speed, and \( r \) is the length of the rope.

Rearrange the Formula to Solve for v

We need to find the maximum speed \( v \) such that the tension does not exceed the maximum allowable tension of 60 Newtons. Rearrange the formula as \( v^{2} = \frac{T \cdot r}{m} \), then solve for \( v \) to get \( v = \sqrt{\frac{T \cdot r}{m}} \).

Plug Parameters into the Formula and Calculate v

Substitute the values for \( T \), \( r \), and \( m \) (user-input) into the rearranged formula: \( v = \sqrt{\frac{60 \times 3}{m}} \). Calculate \( v \) for given values of \( m \).

Compare Results with Allowable Speeds

For each calculated \( v \), check if it matches or exceeds the available speeds (1, 10, 20, 40 m/s) without exceeding the maximum tension limit. The highest matched speed without exceeding the tension is the answer.

Write the Program Logic

Ask for user input for mass \( m \). Calculate maximum allowable speed \( v \) using the formula from Step 2. Compare it to the available speeds and display the maximum allowable speed.

Key concepts

Tension and Forces

When discussing tension and forces, it's critical to understand that tension refers to the force carried by a rope, string, or any flexible connector. In this exercise, the tension is the force exerted on the rope due to the mass being whirled around. The formula given is key:
\[ T = \frac{m v^2}{r} \]
This formula tells us how tension, mass, speed, and radius are related. Tension increases as either the mass or speed increases, or as the radius decreases, under constant conditions.

• Tension \( T \): Measured in Newtons, it is the force exerted along the rope.
• Mass \( m \): This is the weight of the object being spun and is typically measured in kilograms.
• Speed \( v \): This indicates how fast the object is being whirled and is measured in meters per second.
• Radius \( r \): The rope's length, which is 3 meters in this context.

Understanding how these quantities interact can help pinpoint the optimal conditions to prevent the rope from exceeding its tension limit. This context helps illustrate one of the foundational concepts in physics: how forces interact in a dynamic system.

Circular Motion

Circular motion occurs when an object moves in a circular path. This type of motion is common in many physical systems, from carousels at amusement parks to particles in accelerators. In this exercise, the mass attached to the rope is experiencing uniform circular motion.
A key aspect of circular motion is that the object constantly changes direction, which implies a continuous change in velocity. This change equates to acceleration, known as centripetal acceleration. It points towards the center of the circular path.
The tension in the rope plays a crucial role in providing the required centripetal force to keep the mass moving in its circular path. The relation between centripetal force and tension is encapsulated in the formula:

• The centripetal force required is \( F_c = \frac{m v^2}{r} \), which is exactly what the tension \( T \) provides in this scenario.

Thus, the tension must be equal to or greater than this centripetal force for the object to maintain its path without causing the rope to snap. This linkage is what allows us to calculate the maximum speed before the tension exceeds its limit.

Problem Solving in Physics

Problem solving in physics often involves connecting theoretical concepts with practical applications. The exercise demonstrates this by applying the concept of tension in the scenario of a mass whirling on a rope.
One effective approach shown here is to start by identifying the physical principles at play. Recognizing that tension, forces, and circular motion relate directly to the setup is a good start.

• Rearranging Equations: The given formula \( T = \frac{m v^2}{r} \) was manipulated to solve for the unknown speed \( v \). This requires practice in algebra and understanding how variables interplay.
• Parameter Substitution: Next, substituting known values, like maximum tension and rope length, directly into the new equation helps us derive specific solutions.
• Comparative Analysis: Checking which calculated speeds are feasible within the allowed set further refines the solution to account for practical constraints.

These steps exemplify the iterative process students should adopt: define the problem, apply relevant formulas, compute solutions, and evaluate them in the context given. Following a logical series of actions brings theoretical concepts into tangible outcomes.