Question

A \(20 \mathrm{~m}\) rope with mass density of \(0.5 \mathrm{~kg} / \mathrm{m}\) hangs over the edge of a \(10 \mathrm{~m}\) building. How much work is done pulling the rope to the top?

Short answer

The work done is 735 J.

Step-by-step solution

Determine Mass of Rope

The mass density of the rope is given as \(0.5 \mathrm{~kg/m}\). To find the total mass of the rope, we multiply the rope's length by its mass density: \(20 \mathrm{~m} \times 0.5 \mathrm{~kg/m} = 10 \mathrm{~kg}\).

Establish the Work Formula

Work done \(W\) to lift an object is expressed as \(W = F \cdot d\), where \(F\) is the force and \(d\) is the distance the object is lifted. For a variable force along the height, the work required is an integral: \(W = \int F(y) \cdot dy\).

Calculate Force Exerted by Rope

Since the force exerted by the rope at a height \(y\) above the ground is the weight of the section of rope below \(y\), calculate the weight of the rope below \(y\) as \(F(y) = 0.5\mathrm{~kg/m} \times 9.8\mathrm{~m/s^2} \times (20 - y)\).

Set Limits of Integration

As the rope hangs from the edge of a \(10\mathrm{~m}\) building, compute from \(y = 0\,\mathrm{m}\) (building's edge) to \(y = 10\, \mathrm{m}\) (entire rope lifted to top of building).

Integrate to Calculate Work

Substitute \(F(y) = 0.5 \times 9.8 \times (20 - y)\) into the integral and evaluate \(W = \int_{0}^{10} 4.9 (20-y) dy\). Simplifying, we get \(W = \int_{0}^{10} (98 - 4.9y) dy\).

Perform the Integration

Calculate \(\int_{0}^{10} 98\, dy = 98y\bigg|_0^{10}\) and \(\int_{0}^{10} 4.9y\, dy = 4.9 \cdot \frac{y^2}{2}\bigg|_0^{10}\). Evaluate these to get \(98 \times 10 - \frac{4.9 \times 100}{2} = 980 - 245 = 735\).

Present the Final Answer

Thus, the total work done to pull the rope to the top of the building is \(735 \mathrm{~J}\).

Key concepts

Work Done

In integral calculus, the term "work done" refers to the amount of energy required to move an object over a distance when a force is acting upon it. For constant forces, calculating work is straightforward: multiply the force by the distance. However, for forces that change over the distance, such as pulling a rope with varying lengths hanging over a building, integration is necessary.

- **Work Formula for Variable Force**: - When dealing with a variable force, the work done, \( W \), is the integral of the force \( F(y) \) with respect to distance \( y \). - Mathematically, this is represented as \( W = \int F(y) \cdot dy \).

In the scenario of pulling a rope, the force isn't constant. Sections of the rope closer to the top require lifting less rope than those at the bottom. Hence, each incremental piece contributes differently to the total work done.

Variable Force

The concept of variable force is pivotal in understanding how work is done over a distance where the applied force changes. Unlike a fixed force, a variable force changes based on different factors, including distance or time.

- **Understanding Variable Force in the Rope Problem**: - When lifting a rope, the force gradually reduces as more of it is pulled over the edge. - At any height \( y \), the force \( F(y) \) is the weight of the rope below, calculated by multiplying the mass density by gravitational acceleration, and multiplying by the length of the rope below \( y \).Using the formula \( F(y) = 0.5 \times 9.8 \times (20 - y) \), we calculate the force exerted according to the remaining length of the rope that hasn't yet been lifted.

Integration Limits

Integration limits are crucial when calculating work done in scenarios involving variable forces, as they define the range over which the integration occurs. These boundaries specify where to start and stop when finding the total work done across a section of the trajectory.

- **Defining Limits in this Context**: - The exercise specifies the length of the rope hanging over the building, and we lift it from this length up to the building's top. - Hence, the integral limits are from \( y = 0 \) meters (the building's edge where the rope starts) to \( y = 10 \) meters (where the entire rope is at the top).
These integration limits ensure the computation accounts only for the rope length that has influenced the work done.

Mass Density

Mass density is a fundamental concept when discussing objects with uniform mass distribution, such as a rope or wire. It refers to how much mass is contained in a unit length, represented in units of \( \mathrm{kg/m} \).

- **Role in Calculating Work for the Rope**: - The mass density in this exercise is \( 0.5\, \mathrm{kg/m} \). It allows determining the force required at any rope section by providing a constant value that, when multiplied by gravitational acceleration and length, gives the force due to the weight. Computing these forces with variable mass density over the rope's length allows for an accurate calculation of the energy needed to perform the work.