Problem 52 - University Physics with Modern Physics

Question

A uniform \(2.00-\mathrm{m}\) ladder of mass 9.00 \(\mathrm{kg}\) is leaning against a vertical wall while making an angle of \(53.0^{\circ}\) with the floor. A worker pushes the ladder up against the wall until it is vertical. How much work did this person do against gravity?

Step-by-Step Solution

Answer

The work done against gravity is approximately 17.75 J.

Step 1: Understanding Work Done Against Gravity

The work done against gravity is equal to the change in gravitational potential energy of the ladder. Gravitational potential energy (GPE) is calculated using the formula: \[\Delta \mathrm{GPE} = m \cdot g \cdot \Delta h\]where \(m\) is the mass of the ladder, \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)), and \(\Delta h\) is the change in height of the center of mass of the ladder.

Step 2: Calculate Original Height of Center of Mass

Initially, the ladder's center of mass is halfway along its length at an angle of \(53^{\circ}\) to the ground. The initial height \(h_i\) of the center of mass is given by:\[h_i = \frac{L}{2} \cdot \sin(53^{\circ})\]where \(L = 2.00 \, \mathrm{m}\). So,\[h_i = 1.00 \, \mathrm{m} \cdot \sin(53^{\circ}) \approx 1.00 \, \mathrm{m} \cdot 0.7986 = 0.7986 \, \mathrm{m}\]

Step 3: Calculate Final Height of Center of Mass

When the ladder is vertical, the center of mass is directly above the ground at half its length. Thus, the final height \(h_f\) of the center of mass is:\[h_f = \frac{L}{2} = 1.00 \, \text{m}\]

Step 4: Calculate Change in Height of the Center of Mass

The change in height \(\Delta h\) of the ladder's center of mass is the difference between the final and initial heights:\[\Delta h = h_f - h_i = 1.00 \, \text{m} - 0.7986 \, \text{m} = 0.2014 \, \text{m}\]

Step 5: Calculate Work Done Against Gravity

Now, we calculate the work done against gravity using the change in gravitational potential energy:\[\Delta \mathrm{GPE} = m \cdot g \cdot \Delta h = 9.00 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \cdot 0.2014 \, \text{m} \approx 17.75 \, \text{J}\]

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is a form of energy associated with the position of an object in a gravitational field. It's determined by the object's mass, the height of its center of mass relative to a reference point, and the strength of the gravitational field. This energy can be calculated using the formula:

• \[\Delta \mathrm{GPE} = m \cdot g \cdot \Delta h\]
• where \(m\) is the mass, \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\) on Earth), and \(\Delta h\) is the change in height of the object's center of mass.

This concept is crucial when evaluating work done against gravity. For instance, when lifting an object, you increase its gravitational potential energy. Similarly, if an object moves upwards, its gravitational potential energy increases. Understanding this energy conversion helps explain why work must be done to lift objects and how this work is quantified.

Center of Mass

The center of mass of an object is a point where the mass is evenly distributed in all directions. It's essentially the balance point of that object. For uniform objects like a ladder, the center of mass is found at the midpoint if the object has equal density throughout.

• When you know its position, it helps in calculating the potential energy change as you tilt or move the object.
• For the ladder leaning against a wall, the center of mass is initially at half the ladder's height.

As you alter the ladder's position, such as moving it to a vertical stance, the vertical position of the center of mass changes. This change is crucial to calculate how much work is done as you are working against the force of gravity. Essentially, knowing the center of mass allows us to compute how far it is displaced vertically, which forms a key component in energy calculations.

Angle of Inclination

The angle of inclination refers to the angle formed between the base of an object and the horizontal ground. This angle plays a significant role in determining the height of the center of mass when dealing with inclined objects like a ladder.

• If you have an angle, you can determine the object's height above the ground using trigonometric functions, particularly sine.
• For a ladder leaning against a wall, the height of the center of mass when inclined is calculated with:

• Height Formula for Inclined Objects

  \[h_i = \frac{L}{2} \cdot \sin(\text{angle of inclination})\]
• Using the sine function to find the vertical component based on the angle of inclination.

Adjusting the angle of inclination alters the height of the center of mass, affecting the gravitational potential energy and, consequently, the work done against gravity when moving the object. Understanding these principles helps in computing the slight variations in gravitational potential energy encountered when changing the angle of inclination.