Problem 28 - University Physics with Modern Physics

Question

a) If the gravitational potential energy of a 40.0 -kg rock is 500 . J relative to a value of zero on the ground, how high is the rock above the ground? b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

Step-by-Step Solution

Answer

Answer: The rock is placed at a height of approximately 1.27 meters above the ground. If the rock is lifted to twice its original height, its gravitational potential energy would double to 1000 J.

Step 1: Identify the Given Information and Formula

We are given the mass of the rock (m = 40 kg) and its gravitational potential energy (U = 500 J). The gravitational potential energy formula is: U = mgh where U is gravitational potential energy, m is mass of the object, g is the acceleration due to gravity (approximately 9.81 m/s^2), and h is the height of the object above the ground.

Step 2: Solve for Height (h)

Using the formula U = mgh, we can solve for height (h). Rearrange the formula to isolate h: h = U / (mg) Now, plug in the given values to find h: h = (500 J) / (40 kg × 9.81 m/s^2) Calculate the value of h: h ≈ 1.27 m For part a), the rock is approximately 1.27 meters above the ground.

Step 3: Calculate New Potential Energy

For part b), we have to find the new gravitational potential energy if the rock is lifted to twice its original height. If the original height is h, the new height will be 2h.

Step 4: Substitute New Height Back into Formula

Use the formula for gravitational potential energy with the new height (2h): U_new = mg(2h)

Step 5: Calculate New Potential Energy

Since we know the original height (h = 1.27 m), we can calculate the new potential energy at the doubled height: U_new = (40 kg)(9.81 m/s^2)(2 × 1.27 m) Calculate the value of U_new: U_new ≈ 1000 J For part b), the gravitational potential energy of the rock would double to 1000 J if it was lifted to twice its original height.

Key Concepts

Physics Problem Solving

In physics, solving problems involves a clear understanding of both the concepts and the processes required to arrive at the correct answers. This process typically begins by identifying what is given and what is required to find, just like in this exercise where we needed to find the height of a rock above the ground using its gravitational potential energy.
To solve effectively, follow these steps:

• Identify and list all given information in the problem. In our case, this includes the mass of the rock and its gravitational potential energy.
• Identify what the problem is asking for - in this exercise, we needed to find the height and later the new potential energy if the height is doubled.
• Choose the appropriate formula that applies to the problem. For gravitational potential energy, we used the formula involving mass, gravitational acceleration, and height.
• Rearrange the formula to solve for the unknown variable.
• Substitute the given values into the rearranged formula and calculate.

Understanding this process helps not only in solving the problem at hand but also in tackling similar problems by applying a structured approach.

Formulas in Physics

Formulas in physics are tools used to quantify relationships between different physical quantities. They form the backbone of problem-solving in physics by allowing us to calculate unknown variables when certain information is given.
In this exercise, we utilized the formula for gravitational potential energy, which is expressed as:\[ U = mgh \] where:

• \( U \) is the gravitational potential energy, measured in Joules (J).
• \( m \) is the mass of the object, in kilograms (kg).
• \( g \) represents the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface.
• \( h \) is the height of the object above a reference point, measured in meters (m).

To find how high the rock is above the ground, we rearranged the formula to solve for \( h \):\[ h = \frac{U}{mg} \] Plugging in the known values provided the solution, highlighting the power of formulas in determining unknown physical parameters when given sufficient information.

Conservation of Energy

The principle of conservation of energy states that energy in a closed system remains constant, though it can change forms. This principle is crucial in understanding physical processes and solving related problems.
Gravitational potential energy is one form of energy that often changes in scenarios involving objects moving within a gravitational field, such as a rock being lifted. When the rock's height doubles, so does its gravitational potential energy since potential energy in this context is directly proportional to height.
Utilizing the equation \( U = mgh \), a change in any variable directly affects \( U \). In our problem, doubling \( h \) resulted in doubling \( U \), confirming that:

• The relationship \( U \propto h \) showcases the conservation of energy by proving energy is only transformed, not lost, as the potential energy transitionally adjusts to a change in height.
• With the transformation, the system still conserves total energy, reiterating the essence of the conservation of energy: the total amount remains constant, only shifting between kinetic and potential forms as the situation changes.

Understanding this principle aids significantly in resolving complex physics problems by ensuring that energy transformations follow predictable patterns and laws.