Question

A piñata of mass \(3.27 \mathrm{~kg}\) is attached to a string tied to a hook in the ceiling. The length of the string is \(0.810 \mathrm{~m}\), and the piñata is released from rest from an initial position in which the string makes an angle of \(56.5^{\circ}\) with the vertical. What is the work done by gravity by the time the string is in a vertical position for the first time?

Short answer

Answer: The work done by gravity on the piñata is approximately 4.2 Joules.

Step-by-step solution

Calculate the initial height of the piñata

First, we need to find the vertical distance (height) between the initial position of piñata and its lowest position when the string is vertical. To do this, we can use trigonometry. The height can be calculated using the sin function of the given angle (56.5 degrees) and length of the string (0.810 m). h_initial = L * sin(θ) Where L is the length of the string and θ is the angle. Plug in the values, h_initial = 0.810 * sin(56.5) h_initial ≈ 0.679 m

Calculate the final height of the piñata

Since the final position of the piñata is when the string is vertically hanging, the height of the piñata in this position is equal to the length of the string. h_final = L h_final = 0.810 m

Calculate the change in height

Now we can calculate the change in height as the difference between the final height and initial height of the piñata. Δh = h_final - h_initial Δh = 0.810 - 0.679 Δh ≈ 0.131 m

Calculate the work done by gravity

The work done by gravity is the product of the mass of the piñata, the acceleration due to gravity (g ≈ 9.81 m/s^2), and the change in height. W = m * g * Δh Plug in the values, W = 3.27 kg * 9.81 m/s^2 * 0.131 m W ≈ 4.2 J (Joules) The work done by gravity by the time the string is in a vertical position for the first time is approximately 4.2 Joules.

Key concepts

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics dictating that energy cannot be created or destroyed, only transformed from one form to another. This principle is particularly relevant when dealing with mechanical systems like the piñata on a string.

When the piñata is first released, it has potential energy due to its elevated position relative to its lowest possible point of travel. As it swings down towards the vertical, this potential energy is converted into kinetic energy—energy of motion. At the moment the string becomes vertical, if we ignore air resistance and other non-conservative forces, the total mechanical energy (potential plus kinetic) remains constant throughout the piñata's descent.

The conservation of energy allows us to predict the movement and speed without needing to track every interaction along the way—only the initial and final states matter.

Trigonometry in Physics

Trigonometry, the branch of mathematics that deals with triangles and their relationships, is crucial in physics for solving problems that involve angles and distances. This discipline becomes especially useful in analyzing the piñata problem, where we need to find the initial height of the piñata.

Briefly explaining, the string makes an angle with the vertical, forming a right-angle triangle with the string as the hypotenuse. By applying the sine function, we're able to compute the opposite side—the initial height of the piñata above its lowest position. Understanding such trigonometric relationships empowers students to dissect and solve a wide array of physics problems involving angles.

Kinetic and Potential Energy

Kinetic energy is the energy of motion, which any moving object possesses. It's determined by the object's mass and velocity, with the equation given as \( KE = \frac{1}{2}mv^2 \). On the flip side, potential energy is the energy stored in an object due to its position or arrangement. For objects in a gravitational field, like our piñata, this is calculated using \( PE = mgh \), where \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height above a reference point.

In the context of the piñata, its potential energy is highest at the initial release point and diminishes to zero at the lowest swing point, assuming all potential energy converts to kinetic energy. Understanding the exchange between these two types of energy is essential for recognizing how objects move under the influence of gravity, and it gives insights into the work done by gravity on the piñata as it swings.