A box of bananas weighing $40.0$ N rests on a horizontal surface. The coefficient of static friction between the box and the surface is $0.40$, and the coefficient of kinetic friction is $0.20$. What is the magnitude of the friction force if a monkey applies a horizontal force of $6.0$ N to the box and the box is initially at rest?

Verified step by step guidance

1. Determine the maximum static friction force using the formula: $f$_max=μ_sN, where $\mu$_s is the coefficient of static friction and $N$ is the normal force. Here, $N$ equals the weight of the box, which is 40.0 N.
2. Substitute the given values into the formula: $f$_max=0.40×40.0. This will give the maximum static friction force.
3. Compare the applied force (6.0 N) to the maximum static friction force calculated in the previous step. If the applied force is less than or equal to the maximum static friction force, the box will not move, and the friction force will equal the applied force.
4. If the applied force exceeds the maximum static friction force, the box will start moving. In this case, the friction force will transition to kinetic friction, which is calculated using the formula: $f$_k=μ_kN, where $\mu$_k is the coefficient of kinetic friction.
5. Substitute the given values into the kinetic friction formula if needed: $f$_k=0.20×40.0. This will give the kinetic friction force if the box is in motion.

Key Concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It acts when an object is at rest and is proportional to the normal force and the coefficient of static friction. In this scenario, the maximum static friction force can be calculated using the formula: F_static_max = μ_static * N, where μ_static is the coefficient of static friction and N is the normal force.

Normal Force

The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. It acts in the opposite direction to gravity and is equal to the weight of the object when on a horizontal surface. In this case, the normal force acting on the box of bananas is equal to its weight, which is 40.0 N.

Friction Force

The friction force is the force that opposes the relative motion of two surfaces in contact. It can be static or kinetic, depending on whether the surfaces are at rest or sliding past each other. When a horizontal force is applied to the box, the friction force will equal the applied force until it reaches the maximum static friction, which determines whether the box will move or remain at rest.