Question

A load of bricks at a construction site has a mass of \(75.0 \mathrm{~kg}\). A crane with 725 W of power raises this load from the ground to a height of \(45.0 \mathrm{~m}\) at a low constant speed. How long does it take to raise the load?

Short answer

Answer: It takes approximately 45.78 seconds for the crane to raise the load of bricks to the given height.

Step-by-step solution

Find the work done

In order to calculate the work done (W), we will use the equation: $$ W = mgh $$ Where m is the mass of the bricks, g is the gravitational acceleration (approximately \(9.81 \mathrm{~m/s^2}\)), and h is the height. Plugging in the given values, we have: $$ W = (75.0\,\text{kg}) \times (9.81\,\text{m/s}^2) \times (45.0\,\text{m}) $$

Calculate the work done

We can now compute the work done: $$ W = (75.0\,\text{kg}) \times (9.81\,\text{m/s}^2) \times (45.0\,\text{m}) = 33188.75\,\text{J} $$ So, the work done is \(33188.75 \,\text{Joules}\).

Use the power equation to find the time

Now, we will use the power equation: $$ P = \frac{W}{t} $$ Where P is the power (given as \(725\,\text{W}\)) and t is the time we want to find. Rearranging the formula to solve for t, we get: $$ t = \frac{W}{P} $$ Plugging in the values we found in Steps 1 and 2, we have: $$ t = \frac{33188.75\,\text{J}}{725\,\text{W}} $$

Calculate the time

Finally, we can compute the time: $$ t = \frac{33188.75\,\text{J}}{725\,\text{W}} = 45.78\,\text{s} $$ So, it takes approximately \(45.78\, \text{seconds}\) for the crane to raise the load to the given height.

Key concepts

Work-Energy Principle

Understanding the work-energy principle is vital for solving problems in the realm of physics. This principle is a fundamental concept that connects the work done on an object with its energy change. In simple terms, when the work is done on an object, it either gains or loses energy.

Consider our exercise involving a crane lifting a load of bricks. The crane does work by using force to move the bricks over a distance upwards. According to the work-energy principle, this work is converted into potential energy - specifically, gravitational potential energy, as the bricks are lifted against the pull of gravity. This concept allows us to use the equation \( W = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height lifted.

It's essential to remember that this equation assumes that the speed is constant, meaning no kinetic energy is changing - all the work done goes into changing the object's gravitational potential energy. Through this principle, we can predict the energetic outcome of physical processes by knowing the work done.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. It's like the energy stored in a ball held high above the ground before it's dropped.

In our case, the load of bricks gains gravitational potential energy as the crane hoists it up. The higher it goes, the more potential energy it gains. This energy can be calculated by the formula \( U = mgh \), mirroring the work done formula. \( U \) stands for the potential energy, \( m \) for mass, \( g \) for acceleration due to gravity, and \( h \) for height.

It's essential for students to grasp that this potential energy depends on the height from a reference point, which is often chosen to be the ground level for practicality. When the object falls, this stored potential energy is converted back into kinetic energy, unless it's used to do work, as in the case of the crane.

Power-Time Relationship

Power is defined as the rate at which work is done or energy is transferred. When discussing machines like cranes, understanding the power-time relationship is crucial for determining how quickly tasks can be performed.

The power of the crane, given as \( P = 725 \text{W} \), tells us that it can convert that much energy each second. In the context of our exercise, we determined how long it would take the crane to raise the bricks using the relationship \( P = \frac{W}{t} \). Rearranging this, we get \( t = \frac{W}{P} \), which gives us the time by dividing the work done by the crane's power.

There's an inverse relationship between power and time for a given amount of work; more power means less time to perform the same work and vice versa. Understanding this relationship helps in planning and optimizing operations where time efficiency is key.