Problem 60 - College Physics

Question

You are riding in an elevator on the way to the 18 th floor of your dormitory. The elevator is accelerating upward with \(a=1.90 \mathrm{~m} / \mathrm{s}^{2} .\) Beside you is the box containing your new computer; the box and its contents have a total mass of \(28.0 \mathrm{~kg}\). While the elevator is accelerating upward, you push horizontally on the box to slide it at constant speed toward the elevator door. If the coefficient of kinetic friction between the box and the elevator floor is \(\mu_{\mathrm{k}}=0.32,\) what magnitude of force must you apply?

Step-by-Step Solution

Answer

Apply a force of 104.832 N to move the box at constant speed.

Step 1: Identify Forces on the Box

To solve this problem, we need to consider the forces acting on the box in the vertical direction. The forces are the gravitational force, the normal force, and the upward acceleration of the elevator. The gravitational force is given by: \[ F_{ ext{gravity}} = m imes g, \] where \( m = 28.0 \) kg and \( g = 9.8 \) m/s². The upward accelerated force by the elevator is \( F_{ ext{upward}} = m imes a \) where \( a = 1.90 \) m/s².

Step 2: Calculate Total Normal Force

The total normal force is the force exerted by the elevator floor on the box. It is the sum of the gravitational force and the force due to upward acceleration: \[ F_{ ext{normal}} = F_{ ext{gravity}} + F_{ ext{upward}} = m imes g + m imes a. \] Substitute the values: \[ F_{ ext{normal}} = 28.0 imes 9.8 + 28.0 imes 1.90. \]

Step 3: Solve for Normal Force

Calculate the normal force: \[ F_{ ext{normal}} = 28.0 imes 9.8 + 28.0 imes 1.90 = 274.4 + 53.2 = 327.6 \text{ N}. \] This is the force exerted by the elevator floor on the box.

Step 4: Calculate Frictional Force

The frictional force opposing the motion of the box is given by:\[ F_{ ext{friction}} = \mu_k imes F_{ ext{normal}}, \]where \( \mu_k = 0.32 \) is the coefficient of kinetic friction. Substituting the normal force calculated:\[ F_{ ext{friction}} = 0.32 \times 327.6. \]

Step 5: Solve for Frictional Force

Calculate the frictional force: \[ F_{ ext{friction}} = 0.32 \times 327.6 = 104.832 \text{ N}. \] This is the force you must overcome to move the box at a constant speed.

Step 6: Apply Horizontal Force to Move Box

To move the box at a constant speed, you must apply a horizontal force equal in magnitude to the frictional force opposing the box's movement. Therefore, the force you need to apply is: \[ F_{ ext{apply}} = F_{ ext{friction}} = 104.832 \text{ N}. \]

Key Concepts

Newton's Laws of Motion

Understanding Newton's Laws is essential when tackling physics problems involving forces and motion. Newton's First Law of Motion states that an object at rest stays at rest, and an object in motion stays in motion at the same speed and in the same direction, unless acted upon by an external force. This will help us understand why we need to apply force to move an object.
This principle supports the notion of inertia, a property of matter that resists changes in motion. Newton's Second Law of Motion tells us how force relates to motion quantitatively. It states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass, expressed as:
\[ F = m \times a \]
In our elevator problem, we calculate how the normal force increases due to upward acceleration, thus affecting the force needed to push the box. Newton's Third Law of Motion describes action-reaction pairs, where for every action (force) in nature, there is an equal and opposite reaction.

Friction

Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. It is crucial in determining how much force is required to move an object across another surface.
In this problem, the kinetic friction comes into play because the box is sliding. The coefficient of kinetic friction, denoted as \( \mu_k \), quantifies the frictional resistance. The frictional force can be calculated using the normal force and the coefficient of kinetic friction:

• \[ F_{\text{friction}} = \mu_k \times F_{\text{normal}} \]

Here, we calculated \( F_{\text{friction}} \) to find out how much force you need to apply to slide the box towards the elevator door.

Forces

Forces are vectors, meaning they have both magnitude and direction. They cause changes in the speed or direction of an object’s motion. In physics, understanding forces requires knowing what kinds of forces are acting on an object and how they interact.
In our elevator problem, several forces are at play:

• The gravitational force pulling the box downward, which is \( m \times g \), where \( g \) is the acceleration due to gravity.
• The normal force exerted by the elevator floor upwards, combining gravity and the additional force due to the elevator's acceleration.
• Frictional force opposing the box's movement.
• The applied horizontal force needed to move the box at constant speed.

Each force affects the motion or equilibrium of the box in different ways, emphasizing the need to understand vector sums and net forces.

Elevator Physics

Elevator physics can be fascinating as it combines several forces and motion principles, especially Newton's laws, into a real-world everyday scenario.
When the elevator moves, it accelerates either upward or downward. The change in acceleration affects the forces experienced by an object inside. In our problem, the elevator accelerates upwards, and this increases the normal force on the box.
As a result, the frictional force also increases since it is dependent on the normal force. Understanding these dynamics is crucial as they apply to not just elevators, but also amusement park rides, shipping logistics, and other vertical moving systems. By considering both the additional acceleration and standard forces like gravity, you learn how to critically analyze such everyday systems using underlying physics principles.