Question

A grandfather pulls his granddaughter, whose mass is \(21.0 \mathrm{~kg}\) and who is sitting on a swing with ropes of length \(2.50 \mathrm{~m}\), backward and releases her from rest. The speed of the granddaughter at the bottom of the swinging motion is \(3.00 \mathrm{~m} / \mathrm{s}\). What is the angle (in degrees, measured relative to the vertical) from which she is released?

Short answer

To find the angle from which the granddaughter was released on the swing, we first examined her movement using the conservation of mechanical energy (considering the initial potential energy is converted to kinetic energy). After calculating the height difference (using trigonometry) and applying the conservation of mechanical energy principle, we determined that the angle from which the granddaughter is released is approximately \(19.3^{\circ}\) relative to the vertical.

Step-by-step solution

Calculate the initial and final potential energy

First, we need to find the potential energy at the highest point (when she is released) and at the lowest point (when she is at the bottom of the swinging motion). To do this, we'll need to use the formula for potential energy, which is: $$ PE = mgh $$ where: - \(m\) is the mass (21.0 kg) - \(g\) is the acceleration due to gravity (\(9.81 \mathrm{m/s^2}\)) - \(h\) is the height (which can be found using trigonometry) We'll also need the kinetic energy, which is given by the formula: $$ KE = \frac{1}{2}mv^2 $$ where: - \(v\) is the velocity at the bottom of the swinging motion (given as 3.00 m/s)

Calculate the height difference

In order to find the height difference, we first need to find the vertical distance at the bottom of the swing and subtract it from the vertical distance at the top of the swing. To do that, we can use the length of the ropes and the angle (which we'll call \(\theta\)) between the ropes and the vertical axis, as follows: $$ \Delta h = l(1 - \cos{\theta}) $$ where: - \(l\) is the length of the ropes (2.50 m)

Apply the conservation of mechanical energy principle

Since the potential energy at the highest point will be completely converted into kinetic energy at the lowest point, we can write the following equation: $$ PE_{\text{highest point}} = KE_{\text{lowest point}} $$ This translates to: $$ mgh = \frac{1}{2}mv^2 $$ We can then substitute the height difference formula from step 2: $$ mg(l(1 - \cos{\theta})) = \frac{1}{2}mv^2 $$

Solve for the angle

Now, we can solve for the angle, \(\theta\). Since we want to find the angle in degrees, we'll first need to use the cosine inverse function (in radians) and then convert the result to degrees. Canceling the mass \(m\) from both sides and plugging in the given values, we have: $$ g(l(1 - \cos{\theta})) = \frac{1}{2}v^2 $$ Solving for \(\cos{\theta}\): $$ 1 - \cos{\theta} = \frac{v^2}{2gl} $$ $$ \cos{\theta} = 1 - \frac{v^2}{2gl} $$ $$ \theta = \cos^{-1}\left(1 - \frac{v^2}{2gl}\right) $$ Plugging in the values, we have: $$ \theta = \cos^{-1}\left(1 - \frac{(3.00 \mathrm{~m/s})^2}{2 \times 9.81 \mathrm{~m/s}^2 \times 2.50 \mathrm{~m}}\right) $$ Calculating the angle in radians and converting it to degrees: $$ \theta \approx 19.3^{\circ} $$ The angle from which the granddaughter is released is approximately \(19.3^{\circ}\) relative to the vertical.

Key concepts

Potential Energy

Potential energy is the energy stored in an object due to its position in a force field, commonly a gravitational field. For example, when a child is on a swing at the highest point, she has maximal gravitational potential energy, which can be calculated using the formula $$ PE = mgh $$
where m is the object's mass, g is the acceleration due to gravity, and h is the height above a reference point, often the lowest point of the swing. In the context of our exercise, by knowing the mass of the granddaughter and the vertical distance from the highest to the lowest point of the swing, we can calculate her potential energy at the top. It's essential to understand that potential energy is a scalar quantity, meaning it has magnitude but no direction.

Kinetic Energy

Kinetic energy is the energy of motion. It is calculated for any moving object using the equation: $$ KE = \frac{1}{2}mv^2 $$
where m represents mass and v is the velocity of the object. In the swinging motion, kinetic energy is highest at the lowest point because the velocity is maximal there. This energy contributes to the swinging motion allowing the granddaughter to rise again to a certain height on the other side. Kinetic energy is directly proportional to the mass of the object and, more significantly, to the square of its velocity, showing how speed massively influences kinetic energy. Notably, like potential energy, kinetic energy is also a scalar quantity.

Trigonometry in Physics

Trigonometry is a branch of mathematics that studies the relationships between the angles and lengths of triangles. It's often used in physics to resolve forces or calculate distances in scenarios involving angles, such as the swing problem in our exercise. The relevant functions in trigonometry include sine, cosine, and tangent, which relate the angles of a triangle to the ratios of its sides.

In the context of our problem, the cosine function determines the vertical position of the swing given its angular displacement from the vertical. The cosine function is used to express the relationship between the length of the hypotenuse (the rope in this case) and the adjacent side (the vertical height from which the granddaughter was released). The formula $$ \Delta h = l(1 - \cos{\theta}) $$ directly applies trigonometry to find the height difference in the swinging motion, which is essential for calculating potential energy changes.

Angular Displacement

Angular displacement is a measure of the angle through which an object has moved from its initial position. It is commonly measured in radians or degrees. In our swing problem, the granddaughter's angular displacement determines her release point from the vertical.

It's important to understand that linear displacement translates to angular displacement on a circular path. This translation is pivotal in relating the linear motion's speed at the bottom of the swing with the height from which the granddaughter is released. Using trigonometry and the conservation of mechanical energy, we can calculate the angular displacement to find the angle of release, which in our exercise was found to be approximately \(19.3^\circ\). Understanding angular displacement not only helps in solving such problems but also deepens comprehension of rotational motion concepts.