Problem 57 - 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 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

Answer: (a) The work required to wind the entire chain onto the cylinder without any extra weight is 44,145 Joules. (b) The work required to wind the entire chain onto the cylinder with the 50-kg block attached to its end is 58,860 Joules.

Step 1: Calculate the mass of the chain

The chain has a length of 30 meters and a density of 5 kg/m. To find the mass of the chain, we can multiply its density by its length: \(mass_{chain} = density \cdot length\) \(mass_{chain} = 5 \mathrm{kg/m} \times 30 \mathrm{m}\) \(mass_{chain} = 150 \mathrm{kg}\)

Step 2: Calculate the force exerted by gravity

Now let's calculate the force exerted by gravity on the chain. \(force_{gravity} = mass_{chain} \cdot g\) where g = 9.81 \(m/s^2\) is the acceleration due to gravity. \(force_{gravity} = 150 \mathrm{kg} \cdot 9.81 \mathrm{m/s^2}\) \(force_{gravity} = 1471.5 \mathrm{N}\)

Step 3: Calculate the work for part a

To lift the chain, we need to input the same amount of work as the force applied by gravity: \(work = force_{gravity} \cdot distance\) \(work = 1471.5 \mathrm{N} \times 30 \mathrm{m}\) \(work = 44,145 \mathrm{J}\) Answer: a. The work required to wind the entire chain onto the cylinder without any extra weight is 44,145 Joules.

Step 4: Calculate the force for part b

Now let's consider the scenario where we have a 50-kg block attached to the chain. We need to find the force exerted by gravity on the chain with the block to calculate the work required. \(force_{block} = mass_{block} \cdot g\) \(force_{block} = 50 \mathrm{kg} \cdot 9.81 \mathrm{m/s^2}\) \(force_{block} = 490.5 \mathrm{N}\) Next, we find the total force exerted by gravity on the chain and the block: \(force_{total} = force_{gravity} + force_{block}\) \(force_{total} = 1471.5 \mathrm{N} + 490.5 \mathrm{N}\) \(force_{total} = 1962 \mathrm{N}\)

Step 5: Calculate the work for part b

Now we calculate the work required to wind the chain with the block onto the cylinder: \(work = force_{total} \cdot distance\) \(work = 1962 \mathrm{N} \times 30 \mathrm{m}\) \(work = 58,860 \mathrm{J}\) Answer: b. The work required to wind the entire chain onto the cylinder with the 50-kg block attached to its end is 58,860 Joules.

Key Concepts

Force and Motion

Force and motion are fundamental concepts in physics that describe how objects move when subjected to different forces. A force is essentially a push or a pull acting on an object, and it can cause the object to start moving, stop moving, or change its velocity or direction. In this exercise, the main force we consider comes from gravity, acting on the chain and block pulling them downward.

When you lift the chain using the winch, you exert an upward force that counteracts gravity. This action requires work, and the amount of work is determined by the force needed and the distance over which this force is applied. Notably, work is only done when the force causes movement, meaning no movement, no work done.

For understanding work in the context of force and motion, use the formula:

• Work (W) = Force (F) × Distance (d)

The direction of the force is crucial because it must be parallel to the direction of the motion for work to be computed.

Gravity

Gravity is a fundamental force that attracts two bodies towards each other. It's what causes the chain to hang downward from the cylinder in our problem. The Earth's gravity acts on the mass of the chain, creating a force that we call the gravitational force. This force depends on the mass of the object and the acceleration due to gravity, which is approximately 9.81 m/s² on Earth's surface.

To calculate the force of gravity on an object, use the formula:

• Force (gravity) = Mass (m) × Gravitational acceleration (g)

In this problem, the gravitational force on both the chain and the additional block needs to be considered for accurate calculations of work. Thus, understanding gravity as a force that constantly pulls the chain is crucial when calculating the force and subsequently, the work required to lift it.

Mass and Density

Mass is a measure of the amount of matter in an object and is typically measured in kilograms. It's an intrinsic property of the object, meaning it doesn't change regardless of location. Density, on the other hand, is the mass per unit volume of an object. In our exercise, the density of the chain helps us determine its total mass since we know its length and its density.

The relationship between mass, density, and volume is expressed by the formula:

• Mass (m) = Density (ρ) × Volume (V)

Since the chain's volume can be expressed in terms of its length (assuming a uniform cross-section), the mass is straightforward to calculate. This calculation is pivotal for determining how much gravitational force affects the chain and how much work the system requires to lift it.