91Ó°ÊÓ

Newton's Second Law of Motion is a fundamental principle in physics, stating that the force acting on an object is equal to the mass of that object multiplied by its acceleration. This principle can be succinctly expressed using the formula:

• \( F = ma \)

Where:

• \( F \) is the total force applied on the object (measured in newtons)
• \( m \) is the mass of the object (measured in kilograms)
• \( a \) is the acceleration (measured in meters per second squared)

In the coffee cup scenario, the force that prevents the cup from sliding is the static friction force. According to Newton's Second Law, this static friction force is equal to the mass of the cup times its maximum acceleration (\( a_{max} \)). Solving this relationship allows us to determine how fast the car can accelerate without the cup sliding off the roof.

The coefficient of static friction, denoted as \( \mu_{s} \), is a dimensionless value representing the frictional force between two static objects. It indicates how easily one object will start to move over the other under force application.

• This value varies between different pairs of surfaces.
• It depends on both the materials and the surface textures in contact.

In our problem, the coefficient of static friction is given as \( 0.24 \). This means that the frictional force resisting motion (in this case preventing the coffee cup from sliding) is 24% of the normal force pressing the cup against the car roof. The higher the coefficient, the more resistance the surfaces will have against starting motion. By knowing this coefficient and applying it along with Newton's Second Law, we can determine the maximum possible acceleration the car can achieve without causing the cup to slip off.

Acceleration represents how quickly an object changes its velocity over time. It is a vector quantity, having both magnitude and direction. When a car accelerates, it increases its speed.

• Units are typically meters per second squared (\( \, \mathrm{m/s^2} \)).
• Acceleration can also mean slowing down, often referred to as negative acceleration or deceleration.

In the exercise, we calculated the maximum acceleration (\( a_{max}\)) determining how swiftly the car can speed up without the cup sliding. The calculated \( a_{max} \) of approximately \( 2.3544 \, \mathrm{m/s^2} \) represents the threshold. The car maintains grip on the cup up to this point. Any higher acceleration will overcome the static friction, causing the cup to move.

Normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. It is a crucial factor in friction calculations, as frictional force depends directly on it.

• It is equal in magnitude and opposite in direction to the gravitational force acting on the object, when the object is on a horizontal surface.
• Represented as \( N \), the normal force carries units of newtons (\( \, \text{N} \)).

For the coffee cup resting on a car roof, the normal force equals the weight of the cup, which is the product of its mass and the acceleration due to gravity \( g \), i.e., \( N = mg \). The interaction between \( N \) and the coefficient of static friction allows the calculation of static frictional force (\( F_{s} = \mu_{s} \cdot N \)). This frictional force ensures the cup remains stationary relative to the accelerating car, resisting motion until the frictional limit is surpassed.