Problem 20 - University Physics with Modern Physics

Question

In a charming 19 thcentury hotel, an old-style elevator is connected to a counter- weight by a cable that passes over a rotaing disk 2.50 \(\mathrm{m}\) in diameter (Fig. 9.28 ). The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. (a) At bow many rpm must the disk turn to raise the elevator at 25.0 \(\mathrm{cm} / \mathrm{s}\) ? (b) To start the elevator moving, it must be accelerated at \(\frac{1}{8} \mathrm{g}\) . What must be the angular acceleration of the disk, in rad/s'? (c) Through what angle (in radi- ans and degrees) has the disk turned when it has raised the elevator 3.25 \(\mathrm{m}\) between floors?

Step-by-Step Solution

Answer

(a) 1.91 rpm; (b) 0.981 rad/s²; (c) 2.6 radians or 149.09 degrees.

Step 1: Convert Elevator Speed to Angular Speed

To find the rotational speed that corresponds to the linear speed of the elevator, we first convert 25.0 cm/s to meters by recognizing 1 cm = 0.01 m. Thus, the speed is 0.25 m/s. The radius of the disk is half its diameter, or 1.25 m. Using the formula: \[ \omega = \frac{v}{r} \]we have:\[ \omega = \frac{0.25 \ \text{m/s}}{1.25 \ \text{m}} = 0.2 \ \text{rad/s} \]

Step 2: Convert Angular Speed to RPM

Since we need to find the speed in revolutions per minute (rpm), we convert radians per second to rpm:\[ \frac{0.2 \ \text{rad/s}}{2\pi \ \text{radians/rev}} \times 60 \ \text{s/min} = \text{rpm} \]Calculating gives:\[ \frac{0.2 \times 60}{2\pi} \approx 1.909 \ \text{rpm} \]

Step 3: Calculate Angular Acceleration

The elevator must be accelerated at \( \frac{1}{8}g \), where \( g = 9.81 \ \text{m/s}^2 \). This means \( a = \frac{9.81}{8} \). The corresponding angular acceleration \( \alpha \) is given by:\[ \alpha = \frac{a}{r} = \frac{9.81 / 8}{1.25} \approx 0.981 \ \text{rad/s}^2 \]

Step 4: Calculate Angle Turned in Radians

We need the angle \( \theta \) through which the disk turns to raise the elevator 3.25 m. Using the arc length formula:\[ \theta = \frac{s}{r} \], where \( s \) is the distance (3.25 m) and \( r \) is the radius (1.25 m):\[ \theta = \frac{3.25}{1.25} = 2.6 \ \text{radians} \]

Step 5: Convert Radians to Degrees

To convert radians to degrees:\[ 2.6 \times \frac{180}{\pi} \approx 149.086 \ \text{degrees} \]

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates or spins. In this context, it tells us how fast the elevator's wheel is turning to move the cabin up and down. This speed is most commonly measured in radians per second (rad/s) or revolutions per minute (rpm).
To convert a linear speed (like the elevator's speed in meters per second) into angular speed, we use this fundamental relationship:

• Linear Speed ( v ) = the rate at which something moves along a path
• v = r \times \omega , where \( \omega \) is the angular speed and \( r \) is the radius of the rotating object.
• This can be rearranged to find: \( \omega = \frac{v}{r} \)

So, if the elevator travels at 0.25 m/s and the disk’s radius is 1.25 m, we find an angular speed of 0.2 rad/s. This means for every second, the disk turns 0.2 radians, allowing for calculation of movements like how many times it rotates per minute.

Angular Acceleration

Angular acceleration comes into play when an object like the elevator transitions from resting to moving, or changes its speed. Just like straight-line motion has acceleration, so too does circular motion, which is angular acceleration, noted by the symbol \( \alpha \).This describes how quickly the angular speed is changing and is measured in rad/s². For the elevator, you need a certain force to overcome inertia and get it moving. This force is influenced by gravity, denoted as \( g \), which is approximately 9.81 m/s².
To figure out necessary angular acceleration:

• First find the linear acceleration needed: \( a = \frac{1}{8} g = \frac{9.81}{8} \text{ m/s}² \)
• Use the formula: \( \alpha = \frac{a}{r} \)

If \( a = 1.22625 \text{ m/s}² \) and the disk's radius is 1.25 m, substituting these gives an angular acceleration \( \alpha = 0.981 \text{ rad/s}² \). This tells us how much the speedup rate is as the elevator starts moving.

Radian Conversion

Understanding radians is crucial when working with anything that spins or rotates. Radians are a way of expressing angles, and they tell us how far along a circle's arc you've gone.
Radians measure angles based on the circle's radius. For instance, one full circle is \( 2\pi \) radians. So, every radian is related to actual distance: the arc length. When you want to compute how far the elevator moved, you need both the angle \( \theta \) and the circle's radius.
A formula often used is:

• \( \theta = \frac{s}{r} \), where \( s \) is the arc length covered

For example, if the disk raises the elevator 3.25 m, and \( r \) is 1.25 m, \( \theta = 2.6 \) radians. To put this angle in degrees, which may be more intuitive:

• Use the conversion: \( \text{degrees} = \theta \times \frac{180}{\pi} \)

Thus, 2.6 radians convert to about 149.086 degrees, helping visualize how far the elevator moved relative to its full rotation.