Question

A mass \(m=2\) kilograms is attached to the end of a rope of length \(r=3\) meters, The mass is whirled around at high speed. The rope can withstand a maximum tension of \(T=60\) Newtons. Write a program that accepts a rotation speed \(v\) and determines whether such a speed will cause the rope to break. Hint \(T=m v^{2} / r\).

Short answer

Calculate \(T\) using \(T = \frac{2v^2}{3}\) and check if \(T > 60\). If true, output "Break"; otherwise, output "Safe".

Step-by-step solution

Understanding the Problem

We are given a mass attached to a rope and need to determine if a given speed will cause the rope to exceed its maximum tension limit of 60 Newtons. We use the formula for tension in a rotating system: \(T = \frac{m v^2}{r}\).

Rearranging the Formula

We need to use the tension formula properly. Plug in the known values of mass \(m = 2\) kg, and rope length \(r = 3\) meters into the formula. Then calculate \(T\) using these known values.

Programming Logic

Design a function that accepts \(v\), the speed as input. The function should calculate \(T = \frac{m v^2}{r}\) with \(m = 2\) and \(r = 3\).

Comparing Tension to Limit

Compare the calculated tension \(T\) to the maximum tension limit of 60 N. If \(T > 60\), the rope will break; otherwise, it will not.

Final Output Decision

The program should print "Break" if the tension exceeds the limit, or "Safe" if it does not exceed the limit.

Key concepts

Circular motion

In physics, circular motion refers to the movement of an object along a circular path. An important aspect of this type of motion is the centripetal force, which acts towards the center of the circle to keep the object moving in a circular trajectory.
Understanding circular motion is key to solving problems related to objects moving in loops or being rotated, such as a stone tied to a string and whirled around. This situation entails calculating the forces acting on the object, specifically how much tension the string can withstand while the object moves in a circle.

• An object in circular motion constantly changes its direction, which involves acceleration.
• Centripetal force is necessary to maintain this circular path and is given by the formula \(F_{c} = \frac{mv^2}{r}\) where \(m\) is the mass of the object, \(v\) is its velocity, and \(r\) is the radius of the circle.

In our example, the centripetal force is equal to the tension in the string as the mass is swung around. If this tension exceeds certain limits, it can cause the string to break.

Tension in physics

Tension is a type of force, commonly encountered when dealing with objects like ropes or strings. It's the pulling force exerted by each end of an object, such as a rope, that's under stretching.

• In physics problems, tension often appears in situations involving strings or chains.
• The tension must counteract external forces to maintain equilibrium or continue motion.

In the context of the circular motion problem, the tension formula \(T = \frac{mv^2}{r}\) helps us determine if the string can handle the force exerted by the moving mass.
Key factors that affect tension include: the mass of the object, its speed, and the radius of the circle. The higher the speed or the heavier the mass, the greater the tension in the rope will be.
If the calculated tension exceeds the rope's threshold, in this case, 60 Newtons, it indicates that the force is too great, leading to the rope snapping.

Python programming for physics

Python is a versatile and powerful programming language that is increasingly being used in physics to solve problems and model physical systems. When it comes to programming solutions for physics problems, Python shines due to its simplicity and comprehensive set of libraries.
Here's how Python can be applied in the context of our circular motion problem:

• Create functions to handle calculations, such as computing tension \(T = \frac{m v^2}{r}\).
• Use conditional statements to compare computed values against predefined thresholds, like checking if tension exceeds 60 Newtons.
• Implement an interactive program that accepts inputs, such as speed, to dynamically perform calculations.

This practical application not only aids in automating solution processes but also gives you a deeper understanding of physical phenomena by allowing you to simulate and test various scenarios quickly. The outcome of the program—whether the rope breaks or remains safe— provides immediate feedback on the interaction between speed, tension, and physical limits.