# Ball and Bowl and Newton's Second Law

• RIT_Rich
In summary, the conversation discusses the use of Newton's Second Law and conservation of energy to find the kinetic energy of a sphere rolling down a hemisphere. The equations and steps for both methods are explained, with the final answer being gR. It is also noted that the integral of Newton's Second Law is equivalent to conservation of energy.

#### RIT_Rich

Hello everyone.

I'm having a little trouble understanding the whole "ball and bowl" problem using Newton's Second Law. What I'm talking about involves a small shpere of mass m and radius r, rolling down a "bowl" (more like a hemisphere of radius R). How would one go about finding the KE of the sphere at the bottom of the hemisphere using Newton's Second Law?

Thanks a lot

Do you really have to use Newton's second law? If I were doing a problem like this, I would use "conservation of energy".

Newton's law says "force is the rate of change of momentum" (which can actually be taken as a definition of force). In the simple case of constant mass, this is "force= mass times acceleration" or
m dv/dt= F.

Model the cross section of the bowl that includes the ball, as it is rolling down, as a circle of radius R. Let theta be the angle formed by the line from the ball to the center of the circle and then to the bottom of the circle. The gravitational force on the ball is -mg, straight down. The ball does not move straight down because the bowl is applying a force toward its center. The net force is tangent to the bowl. Dividing the gravitational force into components parallel and perpendicular to the tangent we see that the angle between perpendicular and vertical lines is also theta. The net force, tangent to the circle, is -mg sin(theta).

Newton's second law now says m dv/dt= -mg sin(theta) (v is the speed along the circumference of the circle). If we measure theta in radians, then the arclength is given by s= Rtheta so
v= ds/dt= R d theta/dt. Newton's second law is now:
R d2theta/dt20= -g sin(theta).
If we let w= dtheta/dt (the angular velocity), then d2theta/dt2= dw/dt. Using the chain rule, dw/dt= dtheta/dt dw/dtheta= w dw/dtheta. Now the equation is R w dw/dtheta= -g sin(theta). We can rewrite this as w dw= -(g/R) sin(theta) dtheta and integrate. That gives (1/2)w2= (g/R) cos(theta)+ C.

At the top of the bowl theta= pi/2 and cos(pi/2)= 0. Also, since the ball started from rest, w= 0 so we have 0= 0+ C or C= 0.

Our equation is (1/2)w2= (g/R) cos(theta).

At the bottom of the bowl, theta= 0 and cos(0)= 1. Therefore,
(1/2)w2= g/R.

We were asked for kinetic energy at the bottom of the bowl. That is (1/2)m v2= (1/2)(Rw)2= R2((1/2)w2)= R2(g/R)= gR.

Using "conservation of energy", we would have argued that at the top of the bowl, v is 0 and the height is R so the potential energy (relative to the bottom of the bowl) is mgR and the total energy is 0+ mgR= mgR. At the bottom of the bowl, the potential energy is 0 so the kinetic energy must be mgR in order to keep total energy the same. It's that easy!

Actually, if you look at the integral of the equation resulting form Newton's second law, you will see that that IS conservation of energy. If the force, F, depends only on position: F(x), it is a "conservative" force. Newton's second law is m dv/dt= F(x). Again, we can use the chain rule to write this as m v dv/dx= F(x) and rewrite that as m v dv= F(x) dx. Integrating, (1/2)mv2= integral(F(x)dx)+ C or (1/2)mv2- integral(F(x)dx)= C.
The first part of that is kinetic energy. The (negative) integral is the work done against the force: the potential energy. That just says that the total energy is a constant: conservation of energy.

Thanks. Thats the answer I got as well. I just needed to know because it got marked wrong on a webassign HW. I probably just didn't write it the right way...but it had to be right.

Thanks

## 1. What is Newton's Second Law?

Newton's Second Law states that the force acting on an object is equal to its mass multiplied by its acceleration. This means that the heavier an object is, the more force it will require to accelerate, and the faster an object accelerates, the more force it will have.

## 2. How does the Ball and Bowl experiment demonstrate Newton's Second Law?

In the Ball and Bowl experiment, a ball is placed inside a bowl and then the bowl is tilted, causing the ball to roll down the slope. This demonstrates Newton's Second Law because the force of gravity causes the ball to accelerate down the slope, and the mass of the ball determines how quickly it accelerates.

## 3. What is the relationship between force, mass, and acceleration in Newton's Second Law?

The relationship between force, mass, and acceleration in Newton's Second Law is described by the equation F = ma, where F is force, m is mass, and a is acceleration. This means that force is directly proportional to both mass and acceleration.

## 4. How is Newton's Second Law related to inertia?

Inertia is the tendency of an object to resist changes in its motion. Newton's Second Law is related to inertia because it explains that the more mass an object has, the more inertia it will have and the more force will be required to change its motion.

## 5. Can Newton's Second Law be applied to moving objects in a straight line?

Yes, Newton's Second Law can be applied to moving objects in a straight line. In this case, the acceleration is in the same direction as the force, and the equation F = ma still applies. This is commonly seen in situations such as pushing a shopping cart or driving a car in a straight line.