Conservation of Mechanical Energy

[ChaoticLlama]

A small car has an initial speed of 4.0m/s just before it enters a loop. What is the largest value for r (radius) that the loop can have if the car is to remain in contact with the circular track at all times?

Answer: r = 0.327m

What I tried was; before the car enters the loop it posses pure kinetic energy, and at the top of loop it posses pure potential energy.

Therefore..
1/2(m)(v)² = (m)(g)(Δh)
(1/2(v)²) / g = (Δh)
Δh = 0.816m

Since 2r = Δh

Therefore
r = 0.408m

What have I done wrong?

 

[da_willem]

Would the car, at the top , stay in contact with the track without any kinetic energy?

 

[dx]

How much velocity would the car need at the top of the loop to stay in contact with the rails? The question asked you what the largest value of r is so that the car remains in contact with the loop at all times. Try to figure out what conditions would be necessary for the car to stay in contact with the rails. You conpletely ignored this is your attempt. The answer you got is actually the answer to "what is the radius of the loop if the car just reaches the top an falls down?".

 

[ChaoticLlama]

I still do not understand what to do.

Do I need to find the minimum velocity that the car has at the top of the loop? Or is finding the velocity unnecessary, and the radius can be found without it?

 

[lightgrav]

you have to keep the car TOUCHING the track at the top ...
this means find the speed needed at the top (as function of "r").
Then use KE => KE + PE to retain that much KE at the top.

 

[Kenny Lee]

Yea. To keep it just touching, the normal at the top is just about zero.

 

[dx]

Think of gravity. Its pulling the car down. Due to the cars inertia, it exerts a force on the rails. The net force that the rails feel from the car should be >= 0 at the top. But since you want the minimum, you can take it equal to zero. have you learned about circular motion? think of the centrifugal force.

 

[da_willem]

The answer you got is actually the answer to "what is the radius of the loop if the car just reaches the top an falls down?".

Actually, in his answer, the car falls down before reaching the top.