Problem 5 - University Physics with Modern Physics

Question

A 75.0 -kg painter climbs a ladder that is 2.75 \(\mathrm{m}\) long leaning against a vertical wall. The ladder makes a \(30.0^{\circ}\) angle with the wall. (a) How much work does gravity do on the painter? (b) Does the answer to part (a) depend on whether the painter climbs at constant speed or accelerates up the ladder?

Step-by-Step Solution

Answer

(a) Gravity does -1009.125 J of work on the painter. (b) No, the work by gravity does not depend on the rate of climb.

Step 1: Determine the Height Gained by the Painter

First, we need to calculate how high the painter climbs. The painter only moves vertically in the context of gravity's work. Given the ladder's length of 2.75 m and an angle of \(30.0^{\circ}\) with the wall, use the sine function to find the height gained: \[ h = 2.75 \times \sin(30.0^{\circ}) \]Recall that \(\sin(30.0^{\circ}) = 0.5\), so:\[ h = 2.75 \times 0.5 = 1.375 \text{ m} \]

Step 2: Calculate the Gravitational Force on the Painter

The gravitational force acting on the painter is given by the equation:\[ F_g = m \times g \]where \( m = 75.0 \text{ kg} \) is the mass of the painter and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity. Thus:\[ F_g = 75.0 \times 9.8 = 735 \text{ N} \]

Step 3: Compute the Work Done by Gravity

The work done by gravity is given by the equation:\[ W = F \cdot d \cdot \cos(\theta) \]Here, \( F = 735 \text{ N} \) is the gravitational force, \( d = 1.375 \text{ m} \) is the height gained, and \( \theta = 180^{\circ} \) because gravity acts downward while the painter climbs upward. Thus, \( \cos(180^{\circ}) = -1 \).Calculating the work:\[ W = 735 \times 1.375 \times (-1) = -1009.125 \text{ J} \]The negative sign indicates that gravity does negative work as it opposes the motion of the painter climbing up.

Step 4: Consider the Impact of Painter's Speed

The work done by gravity depends solely on the change in vertical height and gravitational force. This is independent of whether the painter climbs with constant speed or accelerates because work calculated from gravitational force and vertical displacement involves no time component.

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics that describes the attractive force acting between two masses. In our painter scenario, it's the force that pulls the painter towards the center of the Earth. Gravity acts vertically downward and is directly proportional to the mass of the object and the acceleration due to gravity.
The gravitational force formula is given by:

• \( F_g = m imes g \)

where:

• \( m \) is the mass of the object (in kilograms), and
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth).

In the example provided, the painter has a mass of 75.0 kg, making the gravitational force acting on him \( 735 \text{ N} \). Understanding gravitational force helps us calculate the work gravity does on an object when it moves in a vertical direction.

Vertical Displacement

Vertical displacement is another important aspect of understanding work done by gravity. It refers to the change in height as an object moves vertically. In the painter's scenario, the vertical displacement is calculated by finding the height he gains while climbing the ladder.
This is done using trigonometric functions applied to the ladder's length and angle with the wall. Specifically, the sine function is used when dealing with right triangles:

• \( h = L \times \sin(\theta) \)

where:

• \( L \) is the ladder's length, and
• \( \theta \) is the angle between the ladder and the wall.

In the exercise, the painter climbs up to a height of 1.375 meters. Knowing the displacement allows us to determine how much work gravity does as it impacts energy change when moving against or with gravitational force.

Constant Speed vs Acceleration

When calculating work done by gravity, it's essential to understand that the concept doesn't depend on how fast or slow the painter climbs the ladder. Whether the painter moves at a constant speed or accelerates, the work done by gravity remains the same.
Here's why:

• Work is defined as the product of force and displacement (\( W = F \cdot d \cdot \cos(\theta) \)).
• The formula doesn't include time or speed directly; it focuses only on vertical displacement and force direction.

Therefore, even if the painter climbs faster (accelerates) or maintains a steady pace (constant speed), the gravitational work, calculated by multiplying gravitational force, vertical displacement, and direction (\( \cos(180^{\circ}) = -1 \)), remains unaffected by his motion style. Understanding this helps learners focus on figuring out displacement and force to assess gravitational work.