Problem 56 - University Physics with Modern Physics

Question

An elevator has mass \(600 \mathrm{kg},\) not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 \(\mathrm{m}\) (five floors) in 16.0 \(\mathrm{s}\) , and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 \(\mathrm{kg}\) .

Step-by-Step Solution

Answer

The maximum number of passengers is 28.

Step 1: Convert Motor Power to Watts

First, convert the motor's power from horsepower to watts. We know that 1 hp is equal to 745.7 watts. Therefore, the motor providing 40 hp will have a power, \(P\), given by:\[P = 40 \text{ hp} \times 745.7 \frac{\text{watts}}{\text{hp}} = 29828 \text{ watts}\]

Step 2: Calculate Work Done by the Elevator

Next, calculate the work done by the elevator to lift itself (and passengers) to a height of 20.0 m. Work done, \(W\), against gravity is calculated as:\[ W = F \cdot d = m \cdot g \cdot h \]where \(m\) is the total mass, \(g\) is the acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\)), and \(h\) is the height. Let \(m_t\) be the total mass and \(h = 20.0 \text{ m}\).

Step 3: Use Power to Determine Maximum Work in 16 Seconds

Power is the rate at which work is done. With the motor power available, the maximum work the motor can perform in 16 seconds is:\[ W_{\text{max}} = P \times t = 29828 \text{ watts} \times 16 \text{ s} = 477248 \text{ J} \]

Step 4: Set Up Equation for Total Passenger Mass

We need the total work equation to equal the maximum work the motor can do. This lets us solve for the total mass:\[ \text{Work done} = (600 + n \times 65) \cdot 9.81 \cdot 20 \]where \(n\) is the number of passengers. Set this equal to \(W_{\text{max}}\):\[ (600 + 65n) \cdot 9.81 \cdot 20 = 477248 \]

Step 5: Solve for Number of Passengers

Solve the equation for \(n\):\[ 196.2(600 + 65n) = 477248 \]Divide both sides by 196.2:\[ 600 + 65n = \frac{477248}{196.2} \approx 2433.72 \]Subtract 600:\[ 65n = 1833.72 \]Divide by 65 to find \(n\):\[ n \approx \frac{1833.72}{65} \approx 28.21 \]Since \(n\) must be a whole number, the maximum number of passengers is 28.

Key Concepts

Work and Energy

In physics, work and energy are closely related concepts that help us understand how forces are used to move objects. Work is done when a force causes an object to move in the direction of the force. The formula for calculating work is given by \( W = F \cdot d \), where \( F \) is the force applied and \( d \) is the distance over which the force is applied. This relationship shows that work is essentially energy transferred to or from an object through the application of force.
\[\text{\[ W = F \cdot d \]} \]
Energy, on the other hand, is the capacity to do work. It can exist in various forms, such as mechanical, thermal, chemical, and electrical. The focus in our exercise is on mechanical energy, which includes kinetic energy (energy of motion) and potential energy (energy due to position, particularly gravitational).

• Kinetic Energy: This is the energy an object possesses due to its motion, calculated by \( KE = \frac{1}{2} m v^2 \).
• Potential Energy: Specifically gravitational potential energy, which can be calculated by \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

In the case of the elevator, the focus is on gravitational potential energy, as the elevator does work against gravity to lift itself and the passengers.

Power and Efficiency

Power is the rate at which work is done, or energy is transferred, over time. It is a measure of how quickly work can be performed. The formula for power is \( P = \frac{W}{t} \), where \( W \) is the work done and \( t \) is the time over which the work is performed. In this exercise, the motor of the elevator has a power rating, indicating how fast it can do work in lifting the elevator and passengers to a certain height.

• In our problem, power is converted from horsepower to watts, a standard SI unit, to match the units needed for calculations in physics.
• Efficiency is sometimes considered in these calculations as well, representing the ratio of useful work output to total energy input, but is not specifically addressed as a concern in this problem.

By understanding power, we can determine how much work the motor can do in a specific amount of time, which is crucial for calculating how many passengers the elevator can carry.

Mechanics

Mechanics is a branch of physics that deals with the motion of bodies under the influence of forces. It is divided into two main subfields: kinematics, the study of motion without considering its causes, and dynamics, the study of motion and the forces that affect it. In our exercise, the elevator is moving vertically upward, which involves dynamics as gravitational forces need to be overcome.
We calculate the work using the mechanics of lifting by setting gravitational force equal to the force exerted by the motor. This allows us to find out how much mass can be moved given the motor's power. In this scenario, the key components calculated include:

• Gravitational force acting on the elevator and passengers.
• The work done to lift both the elevator and passengers.
• The amount of force needed to maintain a constant speed, as this affects energy usage.

By using these mechanics principles, the exercise illustrates how forces are at work in real-world applications like elevators.

Gravitational Force

Gravitational force is an essential concept when dealing with objects in mechanics. It is the force of attraction between two masses. On Earth, this force accelerates objects downwards at \( 9.81 \text{ m/s}^2 \). In the elevator problem, the gravitational force needs to be countered to lift the elevator and its passengers against Earth's pull.

• This force is defined mathematically as \( F = m \cdot g \), where \( m \) is the total mass being lifted and \( g \) is the gravitational acceleration.
• In lifting problems, we use this force in our calculation of work done, as work against gravity is based on the displacement height and the total mass that must be lifted.

The equation that links gravitational force and work done helps us find out how much work is needed to raise the mass of the elevator when filled with passengers. This connection makes it possible to solve for the maximum number of passengers the motor can support while still functioning efficiently.