Question

An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 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 kg.

Short answer

The maximum number of passengers is 28.

Step-by-step solution

Understand the Problem

We need to determine the maximum number of passengers who can ride in an elevator that can ascend 20 meters in 16 seconds. The elevator weighs 600 kg, not including passengers, and the average passenger weighs 65 kg. The motor driving the elevator provides a maximum power output of 40 horsepower.

Convert Horsepower to Watts

Since power is given in horsepower and we need it in Watts, we perform the conversion: 1 horsepower (hp) equals 746 Watts. Thus, 40 hp is equivalent to \( 40 \times 746 = 29840 \text{ Watts} \).

Calculate Work Done by Elevator

The work done is equal to the gravitational force times the height. The formula is \( Work = (m_{elevator} + m_{passengers}) \cdot g \cdot h \), where \( m_{elevator} = 600 \text{ kg} \), \( m_{passengers} = n \times 65 \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( h = 20 \text{ m} \).

Relate Power, Work, and Time

Power is the rate at which work is done. So, we use the equation \( Power = \frac{Work}{Time} \). We have \( Power = 29840 \text{ W} \), \( Time = 16 \text{ s} \). Substitute work from Step 3 and solve for the number of passengers \( n \).

Set Up Equation to Solve for Passengers

Substitute into the power equation: \( 29840 = \frac{(600 + 65n) \times 9.8 \times 20}{16} \). Solve the equation for \( n \).

Solve the Equation

Rearrange and simplify the equation: \( 29840 \times 16 = (600 + 65n) \times 9.8 \times 20 \) leads to \( 477440 = (600 + 65n) \times 196 \) \( 2435.92 = 600 + 65n \) \( 1835.92 = 65n \) \( n \approx 28.24 \). Since \( n \) must be whole number, use \( n = 28 \) as fractional passengers are not possible.

Key concepts

Constant Speed

To solve an elevator physics problem, it is important to understand the concept of constant speed. When an object moves at constant speed, its velocity doesn't change over time. Therefore, any force applied is perfectly balanced by an opposing force. In the case of an elevator, while it ascends, the motor must produce force in such a way that it overcomes gravity but doesn't cause acceleration. This balance ensures a smooth ride for passengers.

Maintaining a constant speed allows us to use straightforward formulas when calculating the work done and power needed. We don't have to worry about additional factors like acceleration or deceleration, which simplifies our calculations. For this reason, constant speed is often assumed in basic physics problems like the one in our exercise to simplify calculations and focus on core concepts.

Power Conversion

In physics, power refers to the rate of doing work or transferring energy over time. For the elevator problem, the motor's power is given in horsepower, a common unit in engines and other mechanical industries. However, in physics calculations involving the International System of Units (SI), we usually convert this to watts.

The conversion between horsepower and watts is essential: 1 horsepower equals 746 watts. This means that for our elevator with a motor capable of 40 horsepower, the power output is converted to 40 times 746 watts, which gives us 29,840 watts. Such conversions help standardize our calculations using the metric system and make it easier to solve problems and compare results across different scenarios.

Work Done

Work done in physics is expressed as the force applied to an object multiplied by the distance over which it is applied. For an elevator moving at constant speed, the work done involves lifting the combined mass of the elevator and its passengers to a certain height. Mathematically, this can be written as:

\[\text{Work Done} = (m_{elevator} + m_{passengers}) \cdot g \cdot h\]

• Here, \( m_{elevator} \) is the mass of the elevator, in this case, 600 kg.
• \( m_{passengers} \) is the total mass of the passengers, calculated as the number of passengers times the average weight per passenger, so \( n \times 65 \) kg.
• \( g \) is the acceleration due to gravity (9.8 m/s²), a constant force.
• \( h \) is the height the elevator travels; in this problem, it's 20 meters.

The full calculation of work done tells us how much energy is needed to lift the elevator and passengers. Understanding this gives insight into how mechanical systems utilize energy to perform their functions.

Passenger Capacity

To determine the maximum number of passengers an elevator can carry, we need to consider both the power output of the motor and the combined mass of the elevator and its passengers.

The provided power output of a motor defines how much work can be performed per unit of time. Given the time, work done, and power limits, we can set up equations to find how many passengers can ride without exceeding the motor's capacity. In our case, we use:
\[\frac{(m_{elevator} + n \times 65) \cdot g \cdot h}{\text{time}} \leq \text{power}\]
This equation ensures that the elevator can operate safely and efficiently without being overburdened. After solving, we find that the integer solution for the maximum number of passengers should be chosen, as fractional passengers are not practical. Thus, understanding the limits established by the power output is crucial in determining safe and efficient passenger capacity for any elevator system.