Problem 2 - College Physics

Question

A stone with a mass of \(0.80 \mathrm{~kg}\) is attached to one end of a string \(0.90 \mathrm{~m}\) long. The string will break if its tension exceeds \(60.0 \mathrm{~N}\). The stone is whirled in a horizontal circle on a frictionless tabletop; the other end of the string remains fixed. (a) Make a free-body diagram of the stone. (b) Find the maximum speed the stone can attain without breaking the string.

Step-by-Step Solution

Answer

The maximum speed the stone can achieve is approximately 8.21 m/s.

Step 1: Create a free-body diagram

In a free-body diagram, represent the stone as a dot. The forces acting on it are the tension \( T \) in the string, pointed towards the center of the circle, and the gravitational force \( mg \), acting downwards. Since the motion is horizontal, gravity doesn't affect the tension.

Step 2: Understanding the Forces

The tension in the string provides the centripetal force required to keep the stone moving in the circle. Thus, the tension \( T \) is equal to the centripetal force \( F_c \), which is given by the formula \( F_c = \frac{mv^2}{r} \), where \( m \) is the mass, \( v \) is the speed, and \( r \) is the radius.

Step 3: Equate Tension to Centripetal Force

Since the tension is the only force providing the centripetal force, we can equate the maximal tension to the centripetal force: \[ T = \frac{mv^2}{r} \]. Here, \( T = 60.0 \text{ N} \), \( m = 0.80 \text{ kg} \), and \( r = 0.90 \text{ m} \).

Step 4: Solve for the Maximum Speed

Rearrange the equation to solve for the speed \( v \):\[ v = \sqrt{\frac{Tr}{m}} \].Substitute the given values: \[ v = \sqrt{\frac{60.0 \times 0.90}{0.80}} \].Calculate the maximum speed.

Step 5: Calculation

Calculate the expression \[ v = \sqrt{\frac{60.0 \times 0.90}{0.80}} = \sqrt{67.5} \approx 8.21 \text{ m/s} \].Thus, the maximum speed the stone can attain without breaking the string is approximately \( 8.21 \text{ m/s} \).

Key Concepts

Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path. When an object is in circular motion, it constantly changes its direction.
To ensure this change, the object requires a constant force directed towards the center of the circle. This is known as the centripetal force.
In our example, this force is provided by the tension in the string.

• The formula for centripetal force is: \[ F_c = \frac{mv^2}{r} \]where \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius of the circle.

Remember, the faster the object moves or the smaller the radius, the greater the centripetal force needed. This is why only a certain speed can be sustained before the string breaks.

Free-body Diagram

A free-body diagram is a tool used in physics to visualize the forces acting on an object. It simplifies complex physical situations, making it easier to understand and solve problems.

In the example of the stone, the free-body diagram will show the stone as a simple dot because we're only interested in the forces.

• The forces include:
  • Tension (\( T \)): points towards the center of the circle.
  • Gravitational Force (\( mg \)): acts downwards but doesn't contribute to horizontal motion.

Creating a free-body diagram helps us ignore unnecessary complexities and focus on the forces directly affecting the stone's motion.

Tension

Tension is the force transmitted through a string, rope, or wire when it is pulled tight by forces acting from opposite ends. It is the force that allows the stone to whirl in the circle.

In this exercise, the string can withstand a maximum tension of 60.0 N.

• Once the tension exceeds this limit, the string will snap.
• Hence, the tension at maximum speed must be equal to this value to prevent breaking.

Tension is a crucial force in many physical systems. Understanding its limits is essential for predicting when a system might fail.

Physics Problem Solving

Solving physics problems often involves breaking the problem into manageable steps. Start by understanding what is being asked and identifying the concepts needed.

Next, use tools like free-body diagrams to visualize the problem. Then, apply relevant physical formulas, like those for centripetal force or tension, to relate known quantities to unknowns.

• In our stone exercise:
  • We used a free-body diagram to isolate forces.
  • We identified the maximum tension as the point of failure.
  • By applying the formulas for centripetal force, we find the maximum speed before breaking.

Systematic problem solving is about building up small pieces of information to arrive at a complete and correct solution.