Hard/Tricky Problem on Energy Conservation

[ShamTheCandle]

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!

 

[ShamTheCandle]

I am sorry if the attachment didn't show up. Here is it.

 

[kerimek]

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.