Problem 22 - University Physics with Modern Physics

Question

You are to design a rotating cylindrical axle to lift 800-N buckets of cement from the ground to a rooftop 78.0 m above the ground. The buckets will be attached to a hook on the free end of a cable that wraps around the rim of the axle; as the axle turns, the buckets will rise. (a) What should the diameter of the axle be in order to raise the buckets at a steady 2.00 cm/s when it is turning at 7.5 rpm? (b) If instead the axle must give the buckets an upward acceleration of 0.400 m/s\(^2\), what should the angular acceleration of the axle be?

Step-by-Step Solution

Answer

(a) The axle diameter should be 0.05 m. (b) The angular acceleration should be approximately 15.75 rad/s².

Step 1: Understanding the Problem

We need to find the diameter of the cylindrical axle that raises buckets at a steady speed and determine the angular acceleration required to lift buckets with a specific upward acceleration.

Step 2: Convert Speed Units

First, convert the given speed from cm/s to m/s. The speed is 2 cm/s; converting gives us \(0.02\) m/s.

Step 3: Calculate the Linear Velocity

With the axle's rotational speed as 7.5 rpm, we convert this to radians per second: \( \omega = \frac{7.5 \times 2\pi}{60} \approx 0.785398 \) rad/s.

Step 4: Relate Linear and Angular Velocity

Use the relationship between linear velocity \(v\) and angular velocity \(\omega\) for a cylindrical axle: \(v = r\omega \). Solve for radius \(r \): \( r = \frac{v}{\omega} = \frac{0.02}{0.785398} \approx 0.0254 \) m.

Step 5: Calculate Diameter

Using the radius found, the diameter \(d\) of the axle is \(2r = 2 \times 0.0254 \approx 0.05 \) m.

Step 6: Determine Angular Acceleration Requirement

When the buckets need an acceleration of 0.400 m/s², use the formula \(a = r\alpha\), where \(a = 0.4\) m/s². Solve for \(\alpha\): \(\alpha = \frac{a}{r} = \frac{0.4}{0.0254} \approx 15.75\) rad/s².

Key Concepts

Angular Velocity

Angular velocity is a key concept when dealing with rotational motion, such as the spinning of a cylindrical axle. It refers to the rate at which an object rotates or spins around a particular axis.
For our exercise, the cylindrical axle must turn at a speed of 7.5 revolutions per minute (rpm). However, to solve rotational motion problems, we often need this in radians per second, as radians are the standard unit of angular measurements. The conversion from rpm to radians per second involves multiplying by \( \frac{2\pi}{60} \), leading us to an angular velocity \( \omega \approx 0.785398 \) rad/s.

• This tells us how fast the axle is rotating.
• Helps in relating rotational speeds to linear motion.
• Useful for designing mechanical systems like our bucket lift.

This conversion is crucial, as it allows the calculation of other values like linear velocity and helps in designing the machinery effectively.

Cylindrical Axle

The cylindrical axle is an essential component of the bucket lifting system in this exercise. Its design directly affects how efficiently we can lift the cement buckets to the desired height. Here are some crucial points to consider:

• It is a cylindrical structure, meaning it has a circular cross-section.
• The cable that lifts the bucket wraps around its rim.
• The diameter of the axle influences the linear velocity experienced by the bucket.

By knowing the desired steady speed of the bucket, which is 0.02 m/s, and the rotational speed, we can determine the necessary diameter by first finding the radius. The formula \( v = r\omega \) gives a direct relationship between linear and angular velocities when it involves a cylindrical axle. Once we calculate the radius, multiplying it by two provides the required diameter, which must be 0.05 m for this task.

Linear Velocity

Linear velocity refers to the speed of an object traveling along a path. In our problem, it is the rate at which the buckets move upwards as the axle rotates. We aim for a linear velocity of 0.02 m/s to ensure a steady upward motion.
This value connects to the rotation of the cylindrical axle through the relationship \( v = r\omega \). This equation tells us:

• The linear speed depends on both the radius of the axle and its angular velocity \( \omega \).
• By controlling the dimensions of the axle and the rotational speed, we can control the linear velocity.
• This allows precise lifting speeds adjusted for safety and efficiency.

Designing machinery with these considerations ensures that the buckets rise smoothly to the rooftop at the targeted speed.

Angular Acceleration

Angular acceleration is crucial when we require changing the rotational speed of an object over time. It is the rate of change of angular velocity and is much like linear acceleration but in rotational terms.
In our problem, we need to determine the angular acceleration \( \alpha \) required to give the buckets an upward acceleration of 0.400 m/s² using the axle. Given the linear acceleration \( a = 0.4 \text{ m/s}^2 \) and the earlier determined radius of 0.0254 m, the formula \( a = r\alpha \) can be used to find \( \alpha \). Solving, we find \( \alpha \approx 15.75 \text{ rad/s}^2 \).

• This ensures that the buckets gain speed as they are lifted.
• Allows applications needing accelerated lifting to design effectively.
• Key in engineering to account for varying lifting needs or machinery agility.

Understanding this concept helps in making the system work more dynamically under different loading conditions.