91Ó°ÊÓ

The coefficient of static friction, often denoted as \( \mu_s \), is a crucial factor in determining how much force is needed to overcome the friction between two surfaces that are at rest relative to each other. It is a dimensionless quantity that represents the ratio of the maximum static frictional force \( f_{s, \text{max}} \) to the normal force \( N \). In this context, the coefficient of static friction between the crate and the floor is 0.37.
To better understand this concept, consider:

• The higher the coefficient of static friction, the greater the force needed to initiate movement.
• This coefficient is typically measured experimentally, as it depends on the materials' properties and surface roughness.

The value of \( \mu_s \) helps us calculate the maximum static frictional force using the formula \[ f_{s, \text{max}} = \mu_s \cdot N \]. In our exercise, this translates into calculating how much force it takes to start moving the crate.

The normal force \( N \) plays a pivotal role in static friction. It is the force exerted by a surface to support the weight of an object resting on it, acting perpendicular to the surface. In our problem, with the crate resting on the floor and the push applied horizontally, the normal force is equal to the gravitational force acting on the crate.
Calculating the normal force involves:

• Finding the weight of the object, which is the product of its mass \( m \) and gravitational acceleration \( g \).
• For the crate, \( N = m \cdot g = 35 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 343 \, \mathrm{N} \).

Understanding the normal force allows us to calculate the static frictional force that opposes motion.

The static frictional force is the force that keeps an object stationary when a force is applied. For stationary objects, this force matches the applied force up to a certain limit, known as the maximum static frictional force \( f_{s, \text{max}} \). If the applied force exceeds this limit, the object will begin to move.
In our crate scenario:

• The maximum static frictional force is determined by multiplying the coefficient of static friction \( \mu_s \) by the normal force \( N \), which is calculated as \[ f_{s, \text{max}} = 0.37 \times 343 = 126.91 \, \mathrm{N} \]. \( f_{s, \text{max}} \) reflects the highest resistance given by static friction before movement initiates.
• The actual static frictional force is equal to the applied force, 110 N, as long as the crate is not moving. Since 110 N is less than \( 126.91 \, \mathrm{N} \), the crate remains stationary.

Newton's Laws of Motion provide the foundation to understand how forces impact the movement of objects. They are especially helpful in problems involving friction. The first and second laws are most applicable here.

• First Law: Inertia - An object remains at rest or moves in a straight line unless acted upon by an external force. In the case of our crate, the static frictional force acts as the external force preventing motion.
• Second Law: Force and Acceleration - The net force acting on an object is equal to the mass of the object multiplied by its acceleration \( F = ma \). This law helps us determine that, when the applied force of 110 N is less than the maximum static frictional force \( f_{s, \text{max}} \), no acceleration takes place, and the object does not move.

These laws guide us in understanding that for any movement to occur, the static frictional force must be overcome by an applied force greater than \( f_{s, \text{max}} \). Therefore, the push of 110 N is insufficient on its own to move the crate without assistance.