# Based on the third law motion of equal and opposite forces, a mass should never accelerate because the net force on it will always be zero. Resolve the issue.

would F=ma answer this question?

If a force is exerted by object A on object B, then B exerts a force of equal magnitude and in opposite direction on A. So, there is then a net force on A causing A to accelrate. But B accelerates in the opposite direction.

But the center of mass of A and B does not accelerate, because the total force exerted on the combined system A plus B is zero.

Try to solve this problem by making clever use of Newton's third Law:

Given a sphere of uniform density rho and radius R, what is the gravitatonal force that one half of the sphere exerts on the other half?

## Well, we certainly have a gravity-ous situation here! Newton's third law states that for every action, there is an equal and opposite reaction. In the case of the gravitational force between the two halves of the sphere, this law holds true as well.

When one half of the sphere exerts a gravitational force on the other half, let's call them "lefty" and "righty," they each experience a force of equal magnitude but in opposite directions. Lefty pulls righty with a certain force, while righty pulls lefty with an equal force.

Now, this might make you think that they cancel each other out, resulting in a net force of zero and no acceleration. But hold your gravity-seekers, we've got a trick up our sleeve called "integration."

When we integrate the countless infinitesimal forces acting between each pair of particles in both halves of the sphere, we discover that the net force on each half affects the center of mass of that respective half. This results in the two halves experiencing a torque that causes them to rotate around their common center of mass.

So, while the total gravitational force on the combined system of both halves is indeed zero, the individual halves of the sphere will experience internal forces that lead to rotation. It's like a cosmic dance party where everyone is grooving in opposite directions!

Now, if only the spheres had a sense of humor, they'd be twirling around with joy!

## Yes, the equation F=ma can help resolve the issue. However, it is important to understand that the net force on an object is not always zero, even if there are equal and opposite forces acting on it. This is because the forces may be acting on different objects or different parts of the same object.

In the case of the third law of motion, if a force is exerted by object A on object B, then according to Newton's third law, object B exerts a force of equal magnitude and in the opposite direction on object A. So, there is indeed a net force on object A, resulting in its acceleration. However, it is important to note that object B also accelerates, but in the opposite direction.

The center of mass of the combined system of objects A and B does not accelerate because the total force exerted on the system is indeed zero. The equal and opposite forces cancel each other out when considering the entire system. But this cancellation of forces does not mean that individual objects within the system do not accelerate.

To solve the problem involving the gravitational force between two halves of a sphere, we can make use of Newton's third law. According to the law, the gravitational force exerted by one half of the sphere on the other half is equal in magnitude and opposite in direction. Therefore, the gravitational force that one half of the sphere exerts on the other half is equal to the gravitational force that the other half exerts on it.

## To solve this problem, we can make clever use of Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.

In this case, we have a sphere of uniform density and radius R, and we want to find the gravitational force that one half of the sphere exerts on the other half.

To get the answer, we can consider the gravitational force between two point masses. The formula for gravitational force between two point masses is given by Newton's law of universal gravitation:

F = (G * m1 * m2) / r^2

Where:

F is the gravitational force between two masses,

G is the gravitational constant (approximately 6.674 × 10^-11 Nm^2/kg^2),

m1 and m2 are the masses of the two objects, and

r is the distance between the centers of the masses.

For our problem, we can consider the two halves of the sphere as two smaller spheres with masses m1 and m2. The radius of each smaller sphere would be R/2.

Since the question asks for the gravitational force that one half of the sphere exerts on the other half, we can assume equal masses for the two halves. Let's call the mass of each half M.

Using the formula above, we can calculate the gravitational force between the two halves:

F = (G * M * M) / (R/2)^2

Simplifying the expression:

F = (4 * G * M^2) / R^2

Therefore, the gravitational force that one half of the sphere exerts on the other half is given by (4 * G * M^2) / R^2, where M is the mass of each half and R is the radius of the sphere.