91Ó°ÊÓ

The normal force is one of the key concepts when analyzing motion, especially in physics problems involving surfaces. It is the support force exerted by a surface perpendicular to an object resting or moving across it. In the example of a baseball player sliding into second base, the normal force (\( F_n \)) balances out the gravitational force as there are no vertical movements involved. This ensures that the player moves horizontally without flying off into the air or sinking into the ground. The formula to calculate the normal force when an object is on a horizontal surface is:

• \( F_n = mg \)

where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \). For our baseball player weighing \( 81 \, \text{kg} \), the normal force is \( 81 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 793.8 \, \text{N} \).
This force acts upward and supports the player as he slides, counteracting the force of gravity pulling him downwards.

Newton's second law of motion is a fundamental principle that connects force, mass, and acceleration, expressed as \( F = ma \). It tells us that the force applied to an object is equal to the mass of the object multiplied by its acceleration. When a player slides into second base, the primary horizontal force he experiences is friction. This frictional force results in a deceleration as it opposes the player's motion.
The frictional force is calculated using the product of the coefficient of kinetic friction and the normal force with the formula:

• \( F_k = \mu_k \times F_n \)

For our example, \( \mu_k = 0.49 \) and \( F_n = 793.8 \, \text{N} \), giving a frictional force of \( 389.962 \, \text{N} \). This force results in deceleration because it acts in the opposite direction of the player's movement.To find the deceleration, we use:

• \( a = \frac{F}{m} \)

Substituting the values:

• \( a = \frac{389.962 \, \text{N}}{81 \, \text{kg}} \approx 4.815 \, \text{m/s}^2 \)

Because the force is reducing his speed, this value is considered negative, representing a deceleration.

Kinematic equations provide powerful tools to analyze motion, particularly when forces like friction come into play. To determine the initial velocity of a player who comes to rest after sliding, we can use the kinematic equation for linear motion:

• \( v = u + at \)

In this formula:

• \( v \) is the final velocity, which is \( 0 \, \text{m/s} \) as the player stops,
• \( u \) is the initial velocity,
• \( a \) is the acceleration (which is negative due to deceleration: \(-4.815 \, \text{m/s}^2\)),
• \( t \) is the time of 1.6 seconds.

To find the initial velocity \( u \), rearrange the equation:

• \( 0 = u - 4.815 \, \text{m/s}^2 \times 1.6 \, \text{s} \)

Solving for \( u \), we find:

• \( u = 4.815 \, \text{m/s}^2 \times 1.6 \, \text{s} = 7.704 \, \text{m/s} \)

This calculation tells us the speed at which the player began his slide onto the base.