Force of Friction between the Sphere and the Floor

Given:
Mass of the iron sphere (m) = 1 kg
Initial velocity of the sphere (u) = 20 m/s
Distance traveled by the sphere (s) = 50 m

To find:
Force of friction between the sphere and the floor (Ff)

Using the kinematic equation:
v^2 = u^2 + 2as

where v is the final velocity and a is the acceleration.

First, let's find the final velocity (v) of the sphere when it comes to rest. Since it comes to rest, the final velocity is 0 m/s.

0 = (20 m/s)^2 + 2a(50 m)
0 = 400 m^2/s^2 + 100a m
a = -400/100 = -4 m/s^2

The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, which is due to the force of friction acting against the motion of the sphere.

Using Newton's second law of motion:
F = ma

where F is the force, m is the mass, and a is the acceleration.

The force of friction (Ff) is given by:
Ff = μN

where μ is the coefficient of friction and N is the normal force.

Since the sphere is on a horizontal surface, the normal force is equal to the weight of the sphere (N = mg).

Let's assume the coefficient of friction between the iron sphere and the cemented floor is μ.

Ff = μN
Ff = μmg

Now, substituting the values:
Ff = μ(1 kg)(9.8 m/s^2)
Ff = 9.8μ N

So, the force of friction between the iron sphere and the floor is 9.8μ Newtons.

Explanation:
- The force of friction between the sphere and the floor causes the sphere to slow down and eventually come to rest.
- The force of friction is directly proportional to the normal force and the coefficient of friction.
- The normal force is the force exerted by the floor on the sphere in the vertical direction and is equal to the weight of the sphere.
- The coefficient of friction represents the roughness or smoothness of the surfaces in contact.
- The negative acceleration indicates that the sphere is decelerating or slowing down.
- The force of friction always acts in the opposite direction to the motion of the object.
- In this case, the force of friction opposes the motion of the sphere, causing it to slow down and eventually come to rest.