Problem 78 - University Physics with Modern Physics

Question

You are designing an elevator for a hospital. The force exerted on a passenger by the floor of the elevator is not to exceed 1.60 times the passenger's weight. The elevator accelerates upward with constant acceleration for a distance of 3.0 \(\mathrm{m}\) and then starts to slow down. What is the maximum speed of the elevator?

Step-by-Step Solution

Answer

The maximum speed of the elevator is approximately 5.94 m/s.

Step 1: Understand the Problem

The force exerted on a passenger by the elevator floor must not exceed 1.6 times their weight. The elevator accelerates over 3 m upward. We need to find the maximum speed it reaches under this constraint.

Step 2: Define the Known Quantities

Let the passenger's weight be denoted by the gravitational force, which is \( F_g = mg \). The maximum force is then \( 1.60 imes F_g = 1.60 imes mg \). The elevator's upward distance is 3 m.

Step 3: Set Up the Equation Using Newton's Second Law

According to Newton's second law, the net force is the total force minus gravitational force, which can be written as \( F = ma \). Substituting the maximum allowable force, \( 1.60mg - mg = ma \). Thus, \( 0.60mg = ma \), giving the acceleration \( a = 0.60g \).

Step 4: Use Kinematics to Find Maximum Speed

We use the kinematic equation \( v^2 = u^2 + 2as \), where \( u = 0 \) (initial speed), \( a = 0.60g \), and \( s = 3 \). Substituting the values gives \( v^2 = 2 imes 0.60 imes 9.8 imes 3 \), leading to the calculation \( v = \sqrt{35.28} \).

Step 5: Calculate and Interpret the Result

Calculate \( v \approx \sqrt{35.28} \approx 5.94 \) m/s. This result is the maximum speed the elevator can reach while keeping the force on a passenger within the required limit.

Key Concepts

Newton's Second Law

Newton's Second Law is one of the cornerstone principles in physics. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In mathematical terms, this principle is expressed as \( F = ma \), where \( F \) is the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration. This equation explains how forces cause changes in motion.
To solve the problem of an elevator's maximum speed while ensuring passenger safety, we apply Newton's Second Law. Here, a key aspect is to manage the forces acting on a passenger, which include the gravitational force and the force exerted by the elevator floor as it accelerates. Knowing these forces helps us determine the acceleration and, subsequently, the maximum speed of the elevator.

Kinematic Equations

Kinematic equations are used to describe the motion of objects without considering the forces that cause the motion. They are essential tools in physics for solving problems involving linear motion with constant acceleration. The primary kinematic equation used here is:
\[ v^2 = u^2 + 2as \]
Where:

• \( v \) is the final velocity,
• \( u \) is the initial velocity,
• \( a \) is the acceleration,
• \( s \) is the distance traveled.

In the context of the elevator problem, the aim is to find the maximum speed \( v \) it can reach.
By inserting the values for acceleration and distance, while knowing that the initial speed \( u \) is zero, we can easily compute the elevator's maximum velocity.

Elevator Dynamics

Elevator dynamics refers to the study of forces and motion specific to elevators. An important component of elevator design is ensuring safety and comfort, achieved by controlling acceleration and speed.
When an elevator accelerates upward, a passenger feels a force due to the acceleration added to the gravitational pull. The net force experienced by a passenger can be described by \( F_{net} = F_{elevator} - mg \), where \( F_{elevator} \) is the force exerted by the elevator floor.
The challenge in this scenario is managing the upward acceleration without the floor exerting a force greater than 1.60 times the passenger's weight. Through Newton's Second Law, we derive the acceleration that ensures this condition is met, thereby influencing the elevator's maximum speed.

Maximum Allowable Force

Understanding the concept of maximum allowable force is crucial in systems that interact with humans, such as elevators. It defines the upper limit of force that ensures safety and comfort without causing any harm or discomfort. In our elevator problem, the maximum allowable force is stipulated as 1.60 times the weight of a passenger.
This constraint necessitates adjustments to acceleration since exceeding this force would make passengers feel heavier and could lead to discomfort or even injury. By using the relationship \( F_{max} = 1.60mg \), we derive a safe acceleration rate and speed. Ultimately, these calculations ensure secure and smooth operation, adhering to prescribed safety standards.