Question

A 35.0 -kg child swings on a rope with a length of \(6.50 \mathrm{m}\) that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of \(4.20 \mathrm{m} / \mathrm{s} .\) What is the tension in the rope?

Short answer

Answer: To find the tension in the rope, we must first calculate the change in height (\(h\)) between the highest and lowest points of the swing using the formula \(h = L - \sqrt{L^2 - \frac{1}{4}m^2v^4 / m^2g^2}\). Then, we can find the radius of the circular path (\(R\)) at the lowest point of the swing, using the equation \(R = \frac{1}{2}mv^2 / mg\). Finally, we can find the tension in the rope using the formula \(T = mg + \frac{mv^2}{R}\), where \(m\) is the mass of the child, \(g\) is the acceleration due to gravity, \(v\) is the speed of the child, and \(R\) is the radius of the circular path. After finding the values for \(h\) and \(R\), we can substitute them into the tension formula and solve for the tension (\(T\)) in the rope.

Step-by-step solution

Identify known values and variables

In this exercise, we are given the mass of the child (\(m = 35.0\,\text{kg}\)), the length of the rope (\(L = 6.50\,\text{m}\)), and the speed of the child at the bottom of the swing (\(v = 4.20\,\text{m/s}\)). We are asked to find the tension in the rope, which we will denote as \(T\).

Calculate gravitational potential energy

At the highest point in the swing, the child has no kinetic energy and maximum potential energy. At the lowest point, they have maximum kinetic energy and the least potential energy. The change in the gravitational potential energy, \(\Delta U\), at the lowest point in the swing can be found using: \(\Delta U = mgh\) where \(m\) is the mass of the child, \(g\) is the acceleration due to gravity (approximately \(9.81\,\text{m/s}^2\)), and \(h\) is the change in height between the highest and lowest points. The change in height can be found using the Pythagorean theorem: \(h = L - \sqrt{L^2 - R^2}\), where \(R\) is the radius of the circular path at the lowest point. At the lowest point, \(R = \frac{1}{2}mv^2 / mg\), so we have: \(h = L - \sqrt{L^2 - \frac{1}{4}m^2v^4 / m^2g^2}\)

Find the net force on the child at the lowest point

At the lowest point in the swing, the net force on the child will be the centripetal force acting towards the center of the circular path, given by: \(F_\text{net} = F_\text{c} = \frac{mv^2}{R}\) The net force on the child is also the difference between the tension in the rope and the gravitational force acting on the child: \(F_\text{net} = T - mg\)

Calculate the tension in the rope

Now we can calculate the tension in the rope by setting the net force on the child equal to the centripetal force: \(T - mg = \frac{mv^2}{R}\) Solving for the tension gives: \(T = mg + \frac{mv^2}{R}\) Using the known values for the mass of the child, the length of the rope, and the speed of the child, we can calculate the tension in the rope: \(T = (35.0\,\text{kg})(9.81\,\text{m/s}^2) + \frac{(35.0\,\text{kg})(4.20\,\text{m/s})^2}{R}\) After finding the change in height \(h\) and the radius of the circular path \(R\) using the formulas in Steps 2 and 3, we can plug in those values and calculate the tension, \(T\), in the rope.