Question

A weight lifter lifts a \(275 \mathrm{kg}\) barbell from the ground to a height of \(2.4 \mathrm{m} .\) How much work has he done in lifting the barbell, and how much work is required to hold the weight at that height? (A) 3234 J and 0 J, respectively (B) 3234 J and 3234 J, respectively (C) 6468 J and 0 J, respectively (D) 6468 J and 6468 J, respectively

Short answer

The correct answer is (C) 6468 J and 0 J, respectively.

Step-by-step solution

Determine the formula for work

The formula to calculate work done is given by the equation: \[ W = F \times d \] where \(W\) is the work, \(F\) is the force applied, and \(d\) is the distance over which the force is applied.

Calculate the force (weight of the barbell)

The force in this case is the weight of the barbell, which can be determined using: \[ F = m \times g \] where \(m\) is the mass of the barbell (275 kg) and \(g\) is the acceleration due to gravity (approximately \(9.8 \mathrm{m/s^2}\)). So, \(F = 275 \times 9.8 = 2695 \mathrm{N} \).

Calculate the work done in lifting the barbell

Now use the formula for work: \[ W = F \times d \] We have already calculated \(F = 2695 \mathrm{N} \) and \(d = 2.4 \mathrm{m} \). So, \[ W = 2695 \times 2.4 = 6468 \mathrm{J} \].

Determine the work required to hold the weight at that height

No additional work is required to hold the barbell at height because the force is not causing any displacement. Therefore, the work done is: \[ 0 \mathrm{J} \].

Compare results with provided answers

Based on the calculations: - Work done in lifting the barbell: 6468 J - Work done in holding the weight: 0 J The correct answer is (C) 6468 J and 0 J, respectively.

Key concepts

work done by a force

When you exert a force on an object, and it moves in the direction of that force, you are doing work on the object. It’s like pushing a grocery cart; if you push it forward and it moves forward, you’re doing work. The amount of work done is given by the equation: \[ W = F \times d \] where \(W\) is the work done, \(F\) is the force applied, and \(d\) is the distance over which the force is applied.
The unit for work is the Joule (J), which is equal to one Newton-meter (N\cdot m).
Let’s consider lifting a barbell. When a weightlifter lifts a barbell, they apply an upward force to overcome gravity. The distance the barbell moves upward gives us the displacement. Hence, \( F \) is the weight of the barbell (force due to gravity) and \( d \) is the height lifted. Specifically, for the exercise:

• Force applied, F = 275 kg \times 9.8 m/s\^2 = 2695 N
• Displacement, d = 2.4 m

Plug these values in: \[ W = 2695 \times 2.4 = 6468 J \].
This means 6468 Joules of work is done to lift the barbell.

gravitational potential energy

Gravitational potential energy (GPE) is the energy an object possesses because of its position in a gravitational field. It's what gives the barbell its 'potential' to do work if it were to fall. The formula for GPE is \[ U = mgh \] where \( U \) is the gravitational potential energy, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height of the object above a reference level.
Returning to our barbell example:

• Mass, m = 275 kg
• Acceleration due to gravity, g = 9.8 m/s\^2
• Height, h = 2.4 m

Substituting in these values: \[ U = 275 \times 9.8 \times 2.4 = 6468 J \]
Which means lifting the barbell increases its gravitational potential energy by 6468 Joules. This energy is 'stored' and could be converted back into kinetic energy if the barbell were dropped.
Interestingly, while the barbell is held steady at that height, no additional work is done despite its potential energy being high, because there's no further displacement.

acceleration due to gravity

Acceleration due to gravity, often denoted as \( g \), is a constant that defines the acceleration of any object freely falling due to gravity. On Earth, \( g \) is approximately \( 9.8 \) meters per second squared (m/s\^2).
It’s essential in calculations involving forces and energies related to gravity, like the weight of an object or its gravitational potential energy.
For instance, in calculating the weight of the barbell, the formula used is: \[ F = m \times g \] where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.
In our example:

• Mass, m = 275 kg
• Acceleration due to gravity, g = 9.8 m/s\^2

So, the weight (force due to gravity) is \[ F = 275 \times 9.8 = 2695 N \]
This weight is the force the weightlifter needs to overcome to lift the barbell. It’s a straightforward calculation but an essential concept in understanding how gravity affects objects on Earth.