Question

A pendulum is made by tying a \(1.33-\mathrm{kg}\) stone to a string \(3.82 \mathrm{~m}\) long. The stone is projected perpendicular to the string, away from the ground, with the string at an angle of \(58.0^{\circ}\) with the vertical. It is observed to have a speed of \(8.12 \mathrm{~m} / \mathrm{s}\) when it passes its lowest point. (a) What was the speed of the stone when projected? (b) What is the largest angle with the vertical that the string will reach during the stone's motion? (c) Using the lowest point of the swing as the zero of gravitational potential energy, calculate the total mechanical energy of the system.

Short answer

The initial speed of the stone is computed using the conservation of momentum, the maximum angle of the pendulum can be determined through the conservation of energy and the total mechanical energy of the system can be calculated with the kinetic energy at the lowest point of the swing.

Step-by-step solution

Conservation of Momentum

In order to find the initial speed of the stone (v), we need to use the conservation of angular momentum. At the point where the pendulum is released, the stone has potential and kinetic energy whereas at the lowest point it only has kinetic energy. Mathematically, it is expressed as: \(mgh + \frac{1}{2}mv_i^2 = \frac{1}{2}mv_f^2\), where m is the mass of the stone, v is the velocity (i for initial and f for final), h is the height of the pendulum above the lowest point and g is the acceleration due to gravity. From the given information, we can calculate h using the trigonometric identity: \(h = l - lcos(\theta)\), where l is the length of the string and \(\theta\) is the initial angle. Substituting these values and solving for \(v_i\) gives the initial speed of the stone.

Conservation of Energy

To find the maximum angle of the pendulum (\(\theta_{max}\)), we use the conservation of energy principle stating that at any point during the motion, the total mechanical energy of the system remains constant. Hence, at the highest point, all the kinetic energy is converted into potential energy. This is represented mathematically: \(mgh_{max} = \frac{1}{2}mv_f^2\), where \(h_{max} = l(1-cos(\theta_{max}))\). By substituting and solving for \(\theta_{max}\), we can find the maximum angle of the pendulum.

Calculation of Total Mechanical Energy

The total mechanical energy of the system can be calculated at any point, since it remains constant during the motion. Using the lowest point as the zero of gravitational potential energy, the total energy is equal to the kinetic energy at that point: \(E = \frac{1}{2}mv_f^2\). By substituting the given values, the total mechanical energy can be calculated.

Key concepts

Conservation of Angular Momentum

Understanding the conservation of angular momentum is critical when analyzing the motion of a pendulum. The principle of conservation of angular momentum states that if no external torque acts on a system, the angular momentum of the system remains constant. A torque is a force that tends to produce rotation.

In the case of our pendulum exercise, when we lift the stone and let it go from a certain height, there is no external torque because only gravitational force acts downwards, which does not cause any change in angular momentum. Here's an interesting aspect: even though the speed of the stone changes as it swings, because the radius (length of string) is also changing throughout the motion, the product of mass, velocity, and the perpendicular distance from the pivot point remains constant.

• At the maximum height, the stone has maximum potential energy and minimum kinetic energy.
• As it falls, potential energy is converted into kinetic energy, increasing the stone's velocity as it descends.
• At the lowest point, the stone has its maximum velocity and hence maximum angular momentum.

This conservation allows us to relate the speed of the stone when it was projected to its speed at the lowest point, helping to solve the first part of the exercise.

Conservation of Energy

Kinetic and Potential Energy in Pendulums

When studying pendulum motion, 'conservation of energy' is a vital concept that interlinks gravitational potential and kinetic energies. As per this principle, the total mechanical energy in an isolated system remains constant if no non-conservative forces (like friction) do work on it.

In the context of the pendulum exercise, the total mechanical energy is conserved throughout the stone's swinging motion. At any point during the pendulum's swing, the sum of its gravitational potential energy and kinetic energy remains unaltered, barring any air resistance or other non-conservative forces.

• When the stone is at its highest, it has the maximum potential energy and zero kinetic energy if we momentarily consider it at rest there.
• When passing through the lowest point, its potential energy is zero (as we've defined this point as the reference zero level), and its kinetic energy is at its maximum due to its highest speed.

Application to the Pendulum Exercise

In our exercise, the conservation of energy enables us to determine the stone's initial and final speeds, as well as to calculate the maximum angle the pendulum will achieve. By equating the total energy at the highest point with that at the lowest, it helps to establish the relationship between height and speed, which is crucial for solving the exercise.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. The higher an object is above a reference point, the greater its gravitational potential energy. GPE is given by the equation: \( GPE = mgh\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height above the reference point.

For the pendulum exercise, gravitational potential energy plays a starring role. As the stone rises, it gains GPE; as it falls, that GPE is transformed into kinetic energy, causing the stone to accelerate. At the highest point in its swing, the pendulum's energy is all potential (assuming it momentarily stops), and at the lowest point, where the GPE is defined as zero, it's all kinetic.

• GPE is at its maximum when the string forms the largest angle with the vertical.
• This energy interchange between GPE and kinetic energy is essential for predicting pendulum behavior such as the stone's speed when launched and its motion's height at various points.

In part (c) of the exercise, determining the GPE at the lowest point simplifies to zero, thus making it easier to estimate the total mechanical energy of the pendulum system.