Centripetal Force, Gravity and Normal

[Biloon]

Homework Statement

Homework Equations

$F_{c} = \frac{mv^{2}}{r}$

The Attempt at a Solution

I think the normal force would be the magnitude of vector sum of centripetal force and gravitational force. So I did:

$N = \sqrt{(\frac{mv^{2}}{r})^{2} + (mg)^{2}}$

However, it isn't one of the answer choice. The actual answer is (c), and I have got no clue why it is (c).

I could imagine why answer is:
(a) Since mg is perpendicular to point Q, normal force is only $\frac{mv^{2}}{r}$
(b) if calculated from the highest tip of the circle...

but where does $2mg$ comes from!?

Thank in advance.

 

[CWatters]

The -2mg part originally comes from the KE converted to PE.

Try writing an equation for the energy of the ball at Q.

 

[Biloon]

Oh thanks, so we approach this problem using conservation of energy.

let v = initial velocity

We have this at Point Q:
KE = KE' + PE..

1/2mv^2 = 1/2mv'^2 + mgr.

mv^2/r - 2mg = mv'^2/r

We also have Force at point Q:
Fn = Fc = mv'^2/r = mv^2/r - 2mg

 

[CWatters]

Yes that how I got the book answer.

 

[HalfLostAndSearching]

I would approach this problem by first understanding the concept of centripetal force, gravity, and normal force. Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle. Gravity is the force of attraction between two objects, and it is always directed towards the center of mass of those objects. Normal force is the force that a surface exerts on an object in contact with it, and it is always perpendicular to the surface.

In this homework problem, it seems that the student is trying to find the magnitude of the normal force acting on an object moving in a circular path. This can be done by considering the forces acting on the object at a specific point on the circle, such as point Q. At this point, the object is experiencing two forces: the centripetal force (F_c) directed towards the center of the circle, and the gravitational force (F_g) directed towards the center of mass of the Earth.

To find the magnitude of the normal force, we can use the formula N = F_c + F_g. However, we need to be careful with the direction of these forces. F_c is directed towards the center of the circle, while F_g is directed towards the center of the Earth. Since these forces are not in the same direction, we cannot simply add them together. Instead, we need to use vector addition to find the magnitude of the normal force.

The correct answer, (c), can be found by using the Pythagorean theorem. We can draw a right triangle with sides F_c and F_g, and the hypotenuse will be the magnitude of the normal force. This is because the normal force is the resultant of these two forces. Using the Pythagorean theorem, we get:

N = \sqrt{F_c^2 + F_g^2}

Substituting F_c = \frac{mv^2}{r} and F_g = mg, we get:

N = \sqrt{(\frac{mv^2}{r})^2 + (mg)^2}

This is the same equation that the student attempted to use, but they forgot to square the gravitational force. This is why their answer is not one of the answer choices.

In conclusion, as a scientist, I would recommend reviewing the concepts of centripetal force, gravity, and normal force, and using vector addition to find the magnitude of the normal force