Question

A rock of mass \(0.773 \mathrm{~kg}\) is hanging from a string of length \(2.45 \mathrm{~m}\) on the Moon, where the gravitational acceleration is a sixth of that on Earth. What is the change in gravitational potential energy of this rock when it is moved so that the angle of the string changes from \(3.31^{\circ}\) to \(14.01^{\circ}\) ? (Both angles are measured relative to the vertical.)

Short answer

Answer: The change in gravitational potential energy of the rock is approximately 0.36 J (Joules).

Step-by-step solution

Identify the formula for gravitational potential energy (PE)

The formula for gravitational potential energy is given by: PE = m * g * h where 'm' is the mass of the rock, 'g' is the gravitational acceleration on the Moon, and 'h' is the height difference between the starting and ending points.

Calculate gravitational acceleration on the Moon

The gravitational acceleration on the Moon is a sixth of that on Earth. The Earth's gravitational acceleration is approximately 9.81 m/s^2, so we can calculate the Moon's acceleration as follows: g_moon = (1/6) * 9.81 \mathrm{m/s}^2 ≈ 1.635 \mathrm{m/s}^2

Determine the height difference

Let's use the given length (L) of the string and the angle changes (angle_initial and angle_final) to find the difference in height (delta_h) using trigonometry. We will use the cosine rule as follows: h_initial = L * (1 - \cos{angle\_initial}) h_final = L * (1 - \cos{angle\_final}) delta_h = h_final - h_initial

Calculate the values for h_initial, h_final, and delta_h

Plugging the given angles and string length into the equations, we get: h_initial ≈ 2.45 * (1 - \cos{3.31^{\circ}}) h_final ≈ 2.45 * (1 - \cos{14.01^{\circ}}) h_initial ≈ 0.0316 \mathrm{m} h_final ≈ 0.3007 \mathrm{m} delta_h ≈ 0.2691 \mathrm{m}

Calculate the change in gravitational potential energy

Now, we just need to plug the calculated values for mass (m), g_moon, and delta_h into the formula for gravitational potential energy to find the change: ΔPE = m * g_moon * delta_h ΔPE ≈ 0.773 \mathrm{kg} * 1.635 \mathrm{m/s}^2 * 0.2691 \mathrm{m} ΔPE ≈ 0.3598 \mathrm{J} The change in gravitational potential energy of the rock is approximately 0.36 J (Joules).

Key concepts

Physics Education

Understanding the principles of physics can sometimes feel overwhelming, but breaking it down into manageable concepts and connecting them to real-world scenarios can help create a lasting foundation of knowledge. In the case of gravitational potential energy, it's a fundamental topic often introduced in the context of work and energy. Learning how to apply the formula for potential energy (\( PE = m \times g \times h \) where 'm' is mass, 'g' is gravitational acceleration, and 'h' is height) can be mastered through practice.

Visual representations, such as scenarios depicting different heights or the use of graphs, can prove invaluable. Furthermore, hands-on experiments or simulations, particularly those that can be performed in less Earth-like conditions such as on the Moon, ignite curiosity and provide tangible understanding. This connection between theoretical knowledge and real-life application is at the heart of physics education, empowering students to understand rather just memorize equations.

Gravitational Acceleration on the Moon

Gravitational acceleration is a critical concept in physics, defining the rate at which an object will accelerate towards the center of a planetary body due to gravity. On Earth, this value is approximately 9.81 meters per second squared, but this value changes depending on the celestial body you are considering. For the Moon, the gravitational acceleration is about one sixth of that on Earth, approximately 1.635 meters per second squared.

This significant difference arises because the Moon is smaller and less massive than Earth, hence its gravitational force is weaker. When solving physics problems, such as computing changes in gravitational potential energy, it’s crucial to adjust the gravitational acceleration value to reflect the environment you’re examining. Without this modification, calculations for scenarios on the Moon (or any other body in our solar system) would be inaccurate.

Trigonometry in Physics

Trigonometry, a branch of mathematics dealing with the relationships between the angles and lengths of triangles, places a pivotal role in physics. It helps us solve problems where the direction and magnitude of forces must be determined, and it's especially useful in scenarios involving vectors, oscillations, and waves.

In our example problem, trigonometry is used to calculate the height of a swinging rock at different string angles. By applying the cosine rule — a basic trigonometric function — we acquire the vertical height component of the swing, which is essential for calculating gravitational potential energy changes. This application shows how mathematical concepts are not isolated theories, but rather key tools for unraveling physical phenomena. Even subtle trigonometric changes can result in a significant transformation in the physical system, underlining the importance of precision in physics.