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Chapter 8: Problem 75

A spherical marble that has a mass of \(50 \mathrm{~g}\) and a radius of \(0.5 \mathrm{~cm}\) rolls without slipping down a loopthe-loop track that has a radius of \(20 \mathrm{~cm}\). The marble starts from rest and just barely clears the loop to emerge on the other side of the track. What is the minimum height that the marble must start from to make it around the loop?

Short Answer

Expert verified
After calculating from the given equation, it can be determined that the marble must start from a minimum height of approximately \(2.5\, m\) (taking into account rounding) in order to just barely clear the loop.

Step by step solution

01

Taking Conservation of Mechanical Energy into account

The law of conservation of mechanical energy states that the total mechanical energy (i.e., the sum of potential and kinetic energy) in any isolated system remains constant provided that no external force is applied. Considering that the marble starts from at rest, its initial potential energy can be represented as \(mgh\), where \(m = 50 \mathrm{~g}\), \(g = 9.81 \mathrm{~m/s^{2}}\) is the acceleration due to gravity, and \(h\) is the minimum height we are trying to find. The final kinetic energy just as the marble is about to complete the loop will be \(\frac{1}{2}mv^{2}\), where \(v\) is the velocity at the top of the loop.
02

Equating gravitational potential energy and kinetic energy

When the marble is at the topmost point of the loop, its kinetic energy should be equal to the sum of its gravitational potential energy and the minimum energy required for it to complete the loop (i.e. the kinetic energy that is equal to the centripetal force). We can balance the energies in the following equation: \(mgh = mg(2r) + \frac{1}{2}mv^{2}\), replacing the value of velocity \(v = \sqrt{gr}\), which is the minimum speed required.
03

Solving for the minimum height

Substitute the known values into the equation and solve for \(h\). After substituting the known values (\(m = 50 \mathrm{~g}\), \(g = 9.81 \mathrm{~m/s^{2}}\), \(r = 20 \mathrm{~cm}\)) and solving the equation for \(h\), the minimum height the marble must start from to make it around the loop can be calculated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mechanical Energy
Mechanical energy is a combination of two main forms of energy: potential energy and kinetic energy. In physics, it allows us to track how energy is transformed in a system. For isolated systems, mechanical energy is conserved, which means the sum of potential and kinetic energy stays constant unless an external force is applied.
  • For instance, in the marble exercise, mechanical energy is conserved when the marble rolls down the track.
  • The initial potential energy of the marble transforms into kinetic energy as it moves.
  • At the highest point of the loop, the energy transitions back into potential energy, reducing the kinetic energy.
This conservation principle is crucial for solving problems that involve motions like the marble's journey around the loop-the-loop.
Potential Energy
Potential energy is the energy stored in an object due to its position or configuration. For gravitational potential energy, this energy is calculated using the formula \[ PE = mgh \] where:
  • \(m\) is mass,
  • \(g\) is the acceleration due to gravity,
  • \(h\) is the height above a reference point.
In the marble's scenario:
  • The initial potential energy is present because the marble is at a certain height above the ground.
  • As the marble descends, this energy is converted into kinetic energy.
  • At the top of the loop, potential energy is once again significant as it helps offset the diminishing kinetic energy.
Understanding potential energy's role aids in predicting how much initial energy is needed to overcome obstacles, like clearing the loop.
Kinetic Energy
Kinetic energy is the energy of motion. When an object moves, it possesses kinetic energy, quantified by the formula: \[ KE = \frac{1}{2}mv^2 \] where:
  • \(m\) is the mass of the object.
  • \(v\) is its velocity.
In the loop-the-loop exercise:
  • The marble gains speed as its potential energy is converted to kinetic energy while descending.
  • At the loop's summit, the marble retains some kinetic energy to ensure it completes the loop without falling off.
  • The concept of kinetic energy is critical to calculating the speed that maintains the marble's path.
Recognizing the interchange between kinetic and potential energy supports us in solving motion-related queries like these and explains why certain speeds are necessary for successful loop completion.

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