91Ó°ÊÓ

Normal force is a force that surfaces exert to support the weight of an object resting on them. It acts perpendicular to the surface of contact. In our problem, the block is placed on a horizontal floor and it's being pushed at an angle, which means we must account for the effect of the vertical component of the applied force as well as the weight of the block when determining the normal force.

To calculate the normal force, we consider two main forces:

• The gravitational force acting downward, which is calculated as the mass of the object times the gravitational acceleration: \( mg \). For this block, it would be \( 3.5 \times 9.8 = 34.3 \text{ N} \).
• The vertical component of the applied force \( F \), which is \( F \sin(\theta) \). This component pushes the block upward and partially offsets the gravitational force, calculated as \( 15 \sin(40^\circ) \approx 9.64 \text{ N} \).

The normal force \( N \) supporting the block would therefore be the gravitational force minus the vertical component of the applied force:\[ N = mg - F \sin(\theta) = 34.3 - 9.64 = 24.66 \text{ N} \]
Understanding normal force is crucial, as it directly affects kinetic friction, impacting how easily the block can be moved across the floor.

Kinetic friction is the resistive force that arises when two surfaces are sliding against each other. It opposes the direction of motion, making it harder for objects to move across surfaces.

The magnitude of kinetic friction \( f_k \) is calculated using the coefficient of kinetic friction \( \mu_k \) and the normal force \( N \). This relationship is expressed by the formula:\[ f_k = \mu_k N \]
In our example, the coefficient of kinetic friction is given as 0.25, and we previously calculated the normal force as approximately 24.66 N, so:\[ f_k = 0.25 \times 24.66 \approx 6.165 \text{ N} \]

This frictional force acts in the opposite direction of the block's motion, reducing the effective horizontal force available to accelerate the block.

Understanding kinetic friction is essential for predicting how different surfaces will interact and how forces will affect an object's motion.

Newton's second law is a fundamental principle of classical mechanics, stating that the acceleration of an object is dependent on the net force acting upon it and inversely proportional to its mass. The law is succinctly expressed in the formula:\[ F = ma \]
Where:

• \( F \) is the net force acting on the object,
• \( m \) is the mass of the object,
• \( a \) is the acceleration produced.

In our problem, the net horizontal force is the horizontal component of the applied force minus the kinetic friction:\[ F_{net} = F \cos(\theta) - f_k = 15 \cos(40^\circ) - 6.165 \approx 5.360 \text{ N} \]
To find the block's acceleration, rearrange Newton's second law to solve for \( a \):\[ a = \frac{F_{net}}{m} = \frac{5.360}{3.5} \approx 1.531 \text{ m/s}^2 \]

This equation tells us how the block speeds up or slows down based on applied forces and it's a crucial tool for analyzing motion in various physical contexts.