Question

A load of bricks at a construction site has a mass of \(75.0 \mathrm{~kg} .\) A crane with \(815 \mathrm{~W}\) of power raises this load from the ground to a certain height in \(52.0 \mathrm{~s}\) at a low constant speed. What is the final height of the load?

Short answer

Answer: The final height of the load is approximately 57.6 meters.

Step-by-step solution

Find the work done (W)

We can find the work done by using the formula for power, \(P = \frac{W}{t}\). Rearranging the formula to find W, we get \(W = P \times t\). Now, substitute the given values for P and t: \(W = (815 \mathrm{~W}) \times (52.0 \mathrm{~s}) = 42380 \mathrm{~J}\)

Find the height (h) using the formula for work done

The formula for work done is \(W = mgh\), where m is the mass, g is the acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)), and h is the height. Rearranging the formula to find h, we get \(h = \frac{W}{mg}\). Now, substitute the given values for W, m, and g: \(h = \frac{42380 \mathrm{~J}}{(75.0 \mathrm{~kg}) \times (9.81 \mathrm{~m/s^2})}\) \(h = \frac{42380}{(75.0) \times (9.81)}\) \(h = \frac{42380}{735.75}\) \(h \approx 57.6 \mathrm{~m}\) So, the final height of the load is approximately 57.6 meters.

Key concepts

Power

Power is a fundamental concept in physics, which refers to the rate at which work is done or energy is transferred or transformed. In simplest terms, power measures how quickly you can accomplish a task or how swiftly energy is used or converted into another form.

For instance, a crane lifting a load at a construction site doesn't perform the task all at once; it does the work over a period of time. Here, power is the work done divided by the time it takes to do it, mathematically expressed as:
\( P = \frac{W}{t} \)
where \( P \) denotes power in watts (W), \( W \) is the work done in joules (J), and \( t \) is the time in seconds (s). The crane with a power output of \( 815 \mathrm{~W} \) lifting the bricks signifies that it's capable of converting 815 joules of energy into mechanical work every second.

Work Done

The concept of 'work done' is tied closely to the idea of using a force to move an object over a distance. If you're transferring energy to an object by moving it, you're doing work on the object. The work done on an object is the product of the force applied to the object and the distance it travels in the direction of the force.

Mathematically, it is expressed as: \( W = F \cdot d \) where \( W \) is work done, \( F \) is the constant force applied, and \( d \) is the displacement in the direction of the force. In the context of the crane and the load of bricks, the work done is equal to the product of the gravitational force (the weight of the bricks) and the height they are lifted, \( h \)—the distance the force moves the object.

Gravitational Potential Energy

Gravitational potential energy (GPE) refers to the energy an object possesses due to its position in a gravitational field. Essentially, the higher an object is above the ground, the more gravitational potential energy it has because of the work that would need to be done against gravity to put it there.

It is quantified by the equation:
\( GPE = mgh \)
where \( m \) is the object's mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point. The amount of work done by the crane to raise the bricks equals the increase in the gravitational potential energy of the bricks. Hence, we use the value of the work done to calculate the height to which the bricks are raised and find the final gravitational potential energy of the raised load.

Acceleration Due to Gravity

Acceleration due to gravity is a constant value that represents the acceleration that an object experiences due to the gravitational pull of the Earth. On the surface of the Earth, this acceleration is approximately \( 9.81 \mathrm{~m/s^2} \).

When an object falls towards the Earth or is thrown into the air, it speeds up or slows down at this rate unless other forces (like air resistance) are significant. This acceleration is crucial when calculating the gravitational force exerted on an object (its weight) using the formula \( F = mg \), where \( F \) is the force in newtons (N), \( m \) is the mass of the object in kilograms (kg), and \( g \) is the acceleration due to gravity. Since work involves a force and a displacement, this acceleration is also an essential factor in determining the work done by lifting an object against Earth's gravity.