Hard/Tricky Problem on Energy Conservation

In summary, the problem involves finding the largest possible value for the radius R, given an initial speed of 4.0 m/s, in order for a car to remain in contact with a circular track at all times. The relevant equations for solving this problem are PE=mgh for potential energy, KE=(1/2)mv² for kinetic energy, and Fc=mv²/R for centripetal force. The given information suggests that the car should not have zero velocity at the top of the circle, and this is an even problem from the "Physics 6th Edition" textbook by Cutnell & Johnson.
  • #1
ShamTheCandle
11
0

Homework Statement


If the car is given an initial speed of 4.0 m/s, what is the largest value that the radius R can have if the car is to remain in contact with the circular tract at all times?
See the attached file (named: car.pdf) for the drawing.

Homework Equations


PE=mgh ----> Formula for Potential Energy
KE=(1/2)mv² ----> Formula for Kinetic Energy
Fc=mv²/R ----> Formula for Centripetal Force

The Attempt at a Solution


My initial attempt was to assume that the car has zero velocity at the top of the circle. However this is not true, since the car should "remain in contact with the circular tract at all times". This is an even problem from my textbook, so no answer is given in the back. By the way, I am referring to "Physics 6th Edition" by Cutnell & Johnson.

Thank you for helping!
 
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  • #2
I am sorry if the attachment didn't show up. Here is it.
 

Attachments

  • car.pdf
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  • #3


I would approach this problem by first defining the physical principles involved. In this case, it is the conservation of energy. This means that the total energy of the system (car) remains constant. We can express this as the sum of the kinetic energy (KE) and potential energy (PE) of the car at any given point on the circular track.

KE = (1/2)mv²
PE = mgh

Where m is the mass of the car, v is its velocity, g is the acceleration due to gravity, and h is the height of the car above the ground.

Next, we need to consider the forces acting on the car. In this case, there are two forces at play - the force of gravity (mg) and the centripetal force (Fc). The centripetal force is the force that keeps the car moving in a circular path. It is given by Fc = mv²/R, where R is the radius of the circular track.

Now, in order for the car to remain in contact with the circular track at all times, the centripetal force must be equal to or greater than the force of gravity. This means that:

Fc ≥ mg

Substituting the expressions for Fc and mg, we get:

mv²/R ≥ mg

Solving for R, we get:

R ≥ v²/g

Therefore, the largest value that the radius R can have is v²/g.

In this problem, we are given the initial velocity of the car as 4.0 m/s. Assuming that the car maintains a constant speed throughout the circular track, the largest value for R would therefore be (4.0)²/9.8 = 1.63 meters.

Note: This solution assumes a frictionless track and neglects air resistance. In reality, these factors would also affect the motion of the car and may result in a different value for the maximum radius.
 

Related to Hard/Tricky Problem on Energy Conservation

1. What is the definition of energy conservation?

Energy conservation refers to the practice of reducing the amount of energy used in order to preserve natural resources and decrease the negative impact on the environment. This can be achieved through various methods such as using energy-efficient appliances, improving insulation in buildings, and reducing energy consumption in transportation.

2. How does energy conservation benefit the environment?

Energy conservation helps to reduce the demand for fossil fuels, which are a major contributor to air and water pollution, as well as greenhouse gas emissions. By conserving energy, we can also help to preserve natural resources and protect wildlife habitats.

3. What are some common barriers to energy conservation?

Some common barriers to energy conservation include high upfront costs for energy-efficient appliances and upgrades, lack of awareness about energy-saving practices, and the convenience and comfort associated with using energy without restriction. In addition, inefficient energy policies and regulations can also hinder conservation efforts.

4. How can individuals play a role in energy conservation?

Individuals can play a significant role in energy conservation by making simple changes in their daily lives, such as turning off lights and unplugging electronics when not in use, using public transportation or carpooling, and choosing energy-efficient products. Educating others about the importance of energy conservation can also have a ripple effect in promoting sustainable behaviors.

5. What are some innovative solutions for energy conservation?

There are many innovative solutions being developed for energy conservation, including smart grid technology, which allows for more efficient distribution of energy, and renewable energy sources like solar and wind power. Additionally, advancements in building design and materials, such as green roofs and energy-efficient windows, can greatly reduce energy consumption in homes and buildings.

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