You and your friends are having a blast swinging on a rope swing and jumping into the lake. When it's your turn, you want to let go of the rope swing when your speed is the fastest. You remember from your physics class that the fastest speed corresponds to the greatest kinetic energy. What location along your swinging path would have the greatest kinetic energy? When you grab the rope and begin to swing, your speed increases until you reach the bottom of the path, where your speed is at a maximum, after which point your speed begins to decrease as the swing takes you up into the air again. Thus, you achieve the greatest speed and kinetic energy at the bottom of your path. The increase in speed you experience as you swing on a rope swing is an example of the conservation of energy for a pendulum. Let's discuss in greater detail the energy of pendulums!
Fig. 1 - A person swinging on a rope swing is an example of a pendulum.
Simple and Physical Pendulums
To begin, let's consider the definition of a pendulum. A pendulum is a system in which an object hangs from a fixed point and oscillates back and forth under the influence of gravity. Gravity acts as the restoring force for the pendulum as it acts pushes the mass toward the equilibrium position. Assuming gravity is the only force acting on the pendulum, the pendulum will oscillate forever until acted upon by another force.
A pendulum is a system in which an object hangs from a fixed point and oscillates back and forth under the influence of gravity.
The two types of pendulums studied in physics are the simple pendulum and the physical pendulum. The physical pendulum is a real pendulum in which the dimensions of the oscillating object are relevant to its motion. In the case of a physical pendulum, the motion is dependent upon the moment of inertia of the pendulum, gravity, and the distance away from the pivot point. The simple pendulum is a pendulum in which we consider the hanging object to be a point mass. The motion of a simple pendulum is independent of the mass of the object and dependent on gravity and the length of the string, which we assume to be massless. As we discuss the conservation of energy in pendulums in this article, we will focus on simple pendulums, so when we refer to a pendulum, we are referring to a simple pendulum.
Energy Changes in a Pendulum
Now, let's discuss the energy of a pendulum. The mechanical energy of an oscillating pendulum includes the kinetic energy \( (K) \) and the potential energy \( (U) \). The conservative force acting on the pendulum that gives potential energy to the system is the force of gravity. Thus, the type of potential energy in the system is gravitational potential energy, which depends on the height of the mass with respect to a chosen zero point. We will call the equilibrium position of the pendulum the zero point, so that the gravitational potential energy is zero at this point. Consider that the mass on a pendulum is lifted so that it is in the position on the right shown in the image below.
Fig. 2 - As a pendulum swings from the right to the left, the potential energy and kinetic energy change with position.
When the pendulum is released from this position, the potential energy decreases until it reaches the equilibrium position, and then increases as it swings up on the other side. On the other hand, since the pendulum is initially at rest, the kinetic energy starts from zero and increases until the equilibrium position, after which point it decreases as the pendulum swings up.
Most of the time, we will assume that the force of air resistance on a pendulum is negligible. If that is the case, the total mechanical energy in the system is constant. In cases where it is not negligible, the air resistance introduces a non-conservative force, meaning the total mechanical energy of the system will decrease as some of the kinetic energy is transformed into other forms of energy, such as heat energy, during oscillation. In this case the pendulum will not oscillate forever, but will decrease in amplitude and mechanical energy until the oscillation eventually stops.
Formula for the Kinetic Energy of a Pendulum
The formula for the kinetic energy, \(K,\) of a pendulum is given by: \[K=\frac{1}{2}mv^2.\,\]In this equation, \(m\) is the mass of the pendulum in kilograms, \(\mathrm{kg},\) and \(v\) is its velocity in meters per second, \(\mathrm{\frac{m}{s}}.\) As mentioned in the previous section, the kinetic energy increases as it moves toward the equilibrium position and decreases as it moves away from the equilibrium position. This is because the kinetic energy is proportional to the square of the velocity of the pendulum. The pendulum begins at rest, and increases in velocity until it passes the equilibrium, at which point the pendulum slows down until it reaches the maximum height, where its velocity is momentarily zero.
Maximum and Minimum Kinetic Energy of a Pendulum
At the equilibrium position of the pendulum, the kinetic energy, and the linear velocity, is at a maximum, as shown in the image below. Since the velocity of the pendulum is zero at the positions of greatest amplitude, the kinetic energy at these positions is also zero. The kinetic energy is never negative, thus these are the locations of the minimum kinetic energy.
Fig. 3 - The kinetic energy of a pendulum has a maximum at the equilibrium position and minima at the locations of greatest amplitude, where the kinetic energy is zero.
Formula for the Potential Energy of a Pendulum
Now, let's discuss the formula for the potential energy, \(U,\) of a pendulum. As mentioned previously, the type of the potential energy in a pendulum system is gravitational potential energy, \(U_{g}.\) So, the formula for the potential energy of a pendulum is: \[\begin{align*}U&=U_g\\[8pt] &=mgh.\,\end{align*}\]In this equation, \(m\) is the mass of the pendulum in kilograms, \(\mathrm{kg},\) \(g\) is the acceleration due to gravity in meters per second squared, \(\mathrm{\frac{m}{s^2}},\) and \(h\) is the greatest height achieved by the pendulum in meters, \(\mathrm{m}.\) As the pendulum swings toward the equilibrium position, the potential energy decreases as the height decreases. The potential energy then increases with height as the pendulum moves away from the equilibrium position.
Maximum and Minimum Potential Energy of a Pendulum
The potential energy of a pendulum has maxima at the locations where the pendulum achieves the greatest height, as shown in the image below. Since we have defined the equilibrium position to be the zero point, the height, and thus the potential energy, of the pendulum is zero at this location.
Fig. 4 - The potential energy of a pendulum is greatest at the locations of greatest amplitude, and has a minimum at the equilibrium position, where the potential energy is zero.
Conservation of Energy in a Pendulum
If the force of air resistance on the pendulum is negligible, the total mechanical energy in the system is conserved. This means that the change in mechanical energy as the pendulum moves from one position to another position is zero, or in other words, the mechanical energy is constant. The conservation of energy in a pendulum can be described by this equation: \[\Delta E=\Delta K+\Delta U=0.\]
When other forces, such as air resistance, act on a pendulum, we must consider the dissipated energy in the equation for the conservation of energy as well. There is a decrease in the mechanical energy as some of the kinetic energy is dissipated as heat energy. When this occurs, there is a change in the internal energy, \(IE\), of the system which must be accounted for. Then the equation to describe the conservation of energy in a pendulum is: \[\Delta E=\Delta K+\Delta U+\Delta IE=0.\]
A \(0.5\,\mathrm{kg}\) mass is swinging back and forth on a string of length \(0.5\,\mathrm{m}.\) At the maximum height, the string makes an angle of \(25^{\circ}\) with respect to the vertical. Find the kinetic energy and the velocity of the pendulum when it's at the equilibrium position. Ignore air resistance.
Fig. 5 - The mechanical energy of a pendulum at the highest point and lowest point is constant.
Let's consider the total mechanical energy of the system at the maximum height and at the equilibrium position. At the maximum height, the mechanical energy is the sum of the kinetic and potential energies: \(E_1=K_1+U_1.\) As mentioned previously, the kinetic energy at this location is zero, \(K_1=0,\) so that \(E_1=U_1.\) Substituting in the equation for the gravitational potential energy, we get: \[\begin{align*}E_1&=U_1\\[8pt]&=mgh.\end{align*}\] We can write the height, \(h,\) in this equation in terms of the length of the string and the angle the string is from the vertical using trigonometry so that \(h=L-L\cos\theta\\[8pt]=L(1-\cos\theta).\,\)Then, we have: \[E_1=mgL(1-\cos\theta).\,\]At the equilibrium position, we can write the total mechanical energy as: \(E_2=K_2+U_2.\) The height with respect to the zero point at this location is zero, so \(U_2=0.\) Thus, we can write: \[\begin{align*}E_2&=K_2\\[8pt]&=\frac{1}{2}mv^2,\end{align*}\]where \(v\) is the velocity of the pendulum at the equilibrium position.
The law of the conservation of energy tells us that \(\Delta E=0,\) so we can write: \[\begin{align*}\Delta E&=E_2-E_1\\[8pt]&=0\\[8pt]E_2&=E_1\\[8pt]K_2&=U_1.\end{align*}\]Thus, we see that the kinetic energy of the pendulum at the equilibrium location is equivalent to the potential energy at the maximum height. Let's solve for it now! \[\begin{align*}K_2&=U_1\\[8pt]&=mgL(1-\cos\theta)\\[8pt]&=(0.5\,\mathrm{kg})\left(9.8\,\mathrm{\frac{m}{s^2}}\right)(0.5\,\mathrm{m})(1-\cos(25^{\circ}))\\[8pt]&=0.23\,\mathrm{J}.\,\end{align*}\]Now, let's solve for the velocity: \[\begin{align*}K_2&=\frac{1}{2}mv^2\\[8pt]v^2&=\frac{2K_2}{m}\\[8pt]v &=\sqrt{\frac{2K_2}{m}}\\[8pt]&=\sqrt{\frac{2(0.23\,\mathrm{J})}{0.5\,\mathrm{kg}}}\\[8pt]&=0.96\,\mathrm{\frac{m}{s}}.\end{align*}\]
Energy in Pendulum - Key takeaways
- A pendulum is a system in which an object hangs from a fixed point and oscillates back and forth under the influence of gravity.
- If a pendulum is oscillating only under the influence of gravity, the total mechanical energy of the system is conserved, and the pendulum will oscillate until acted upon by another force.
- If an oscillating pendulum is in motion under the influence of gravity and air resistance, the total mechanical energy of the system is not conserved, and the pendulum will decrease in oscillation until it stops.
- The kinetic energy of a pendulum is maximized at the equilibrium position and minimized at the position of greatest height.
- The potential energy of a pendulum is maximized at the position of greatest height and minimized at the equilibrium position.
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