Problem 55 - Calculus Early Transcendentals

Question

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of \(5 \mathrm{kg} / \mathrm{m}\). a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a \(50-\mathrm{kg}\) block is attached to the end of the chain?

Step-by-Step Solution

Answer

Short Answer: a) Without the block, the work required is \(22050\,\text{J}\). b) With the 50-kg block attached to the end of the chain, the work required is \(36750\,\text{J}\).

Step 1: Calculate the total mass of the chain

Since the chain has a constant density of \(5 kg/m\), we can find its total mass by multiplying density with its length: $$ m_{chain} = \rho \times L = 5\,\text{kg/m} \times 30\,\text{m} = 150\,\text{kg} $$

Step 2: Calculate the potential energy of the chain

As the chain is being wound, its center of mass is being raised. The potential energy of an object is given by: $$ PE = mgh $$ where m is the mass, g is the acceleration due to gravity (around \(9.8 m/s^2\)), and h is the height. The average height the chain is raised is half of its length, \(h = 15\,\text{m}\). Now, calculate the potential energy change: $$ \Delta PE_{chain} = m_{chain} \times g \times h = 150\,\text{kg} \times 9.8\,\text{m/s}^2 \times 15\,\text{m} $$

Step 3: Calculate the work required to wind the chain

The work required to wind the chain is equal to the change in potential energy of the chain: $$ W_{chain} = \Delta PE_{chain} = 150\,\text{kg} \times 9.8\,\text{m/s}^2 \times 15\,\text{m} = 22050\,\text{J} $$ #b. The work to wind the entire chain with the 50-kg block attached#

Step 4: Calculate the potential energy change in the block

The 50-kg block also has potential energy, and its entire potential energy change is given by: $$ \Delta PE_{block} = m_{block} \times g \times h_{block} = 50\,\text{kg} \times 9.8\,\text{m/s}^2 \times 30\,\text{m} $$

Step 5: Calculate the total work required to wind the chain and the block

Now, just add the potential energy changes of the chain and the block to find the total work required: $$ W_{total} = \Delta PE_{chain} + \Delta PE_{block} = (150\,\text{kg} \times 9.8\,\text{m/s}^2 \times 15\,\text{m}) + (50\,\text{kg} \times 9.8\,\text{m/s}^2 \times 30\,\text{m}) = 22050\,\text{J} + 14700\,\text{J} = 36750\,\text{J} $$ The work required to wind the entire chain onto the cylinder using the winch is \(22050\,\text{J}\) without the block, and \(36750\,\text{J}\) with the 50-kg block attached to the end of the chain.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position or height. It plays a significant role in the work-energy principle, which states that work done on an object results in a change in its energy. When winding the chain, both the chain and any attached block are lifted against gravity. This increases their potential energy.
The formula for gravitational potential energy is given by: \( PE = mgh \), where:

• \( PE \) is potential energy
• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\)), and
• \( h \) is the height above the reference level.

Understanding potential energy helps us appreciate how energy conservation works, as the winding of the chain translates to converting potential energy into mechanical work needed.

Density

Density is a measure of mass per unit volume. In this particular exercise, the chain has a constant density, denoted as \( \rho \). This allows us to calculate the total mass of the chain, which is crucial for understanding the potential energy involved when the chain is lifted.
The formula to obtain the mass from density is: \[ m = \rho \times V \] In the case of a one-dimensional problem like this chain, we can think of volume in terms of length, so the formula becomes: \[ m = \rho \times L \] where \( m \) is the mass, \( \rho \) is the density \(5 \text{ kg/m} \), and \( L \) is the length of the chain \(30 \text{ m} \).
By applying these principles, the mass of objects in physical scenarios can be efficiently calculated from given density.

Mass Calculation

When dealing with problems involving work and energy in physics, knowing the mass of the objects involved is essential. Mass is integral for calculations related to potential energy, among other physical properties.
For this chain, we calculated its mass using: \[ m_{chain} = \rho \times L = 5\,\text{kg/m} \times 30\,\text{m} = 150\,\text{kg} \]Understanding mass calculation helps simplify complex systems by focusing on their essential components. For example, knowing the mass of the chain allows us to determine the amount of work involved in raising it. In this exercise, figuring out the mass of the block, which is a given 50 kg, is straightforwardly integrated into the total work required calculations.

Gravitational Force

Gravitational force is the attractive force that the Earth exerts on objects, pulling them toward its center. This concept is key when calculating work and potential energy changes because it defines the energy required to move an object in a gravitational field.
The force due to gravity on an object is given by: \[ F_g = mg \] where:

• \( F_g \) is the gravitational force
• \( m \) is the mass
• \( g \) is the acceleration due to gravity \(9.8 \text{ m/s}^2\)

An object's weight is a manifestation of gravitational force and can be thought of as this force compressing the object towards the Earth. By understanding gravitational force, one can accurately assess the energy needed to perform tasks such as lifting a chain or block against gravitational pull.