At what position of the pendulum is the potential energy maximum What is the kinetic energy at this point?

1. At what point in the pendulum is the potential energy maximum?

As an object falls under the influence of gravity, potential energy is greater than kinetic energy after halfway point/ before the halfway point.

As the pendulum falls to its lowest point, its potential energy is converted into kinetic energy. This is because as the pendulum falls to its lowest point, it speeds up more and more. Thus, at its lowest point, the pendulum has its kinetic energy at a maximum

2. At what point in the pendulum is the kinetic energy zero?

Lowest point

An active pendulum has the most kinetic energy at the lowest point of its swing when the weight is moving fastest. An ideal pendulum system always contains a stable amount of mechanical energy, that is, the total of kinetic plus potential energy.

The potential energy of the pendulum is 0 when the pendulum is at its equilibrium position.

3. At what point in the pendulum is the potential energy zero?

The potential energy of the pendulum is 0 when the pendulum is at its equilibrium position. Therefore, at this point, the mechanical energy E is equal to the kinetic energy KE (all the energy at the equilibrium position is kinetic).

4. At what point in the pendulum is the kinetic energy maximums?

As the pendulum falls to its lowest point, its potential energy is converted into kinetic energy. This is because as the pendulum falls to its lowest point, it speeds up more and more. Thus, at its lowest point, the pendulum has its kinetic energy at a maximum.

5. What is the total mechanical energy of the pendulurn?

Ignoring friction and other non-conservative forces, we find that in a simple pendulum, mechanical energy is conserved. The kinetic energy would be KE= ½mv2,where m is the mass of the pendulum, and v is the speed of the pendulum.

Kinetic energy is energy of motion. At every point in the motion of the pendulum the total mechanical energy is conserved. The sum of the gravitational potential energy and the kinetic energy, at each point of the motion, is a constant, which is the total mechanical energy.

6. Describe the Law of Conservation of Mechanical Energy.

Law of Conservation of Mechanical Energy: The total amount of mechanical energy, in a closed system in the absence of dissipative forces (e.g. friction, air resistance), remains constant. This means that potential energy can become kinetic energy, or vice versa, but energy cannot “disappear”.

In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. For instance, chemical energy is converted to kinetic energy when a stick of dynamite explodes.

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Energy in a Pendulum

In a simple pendulum with no friction, mechanical energy is conserved.  Total mechanical energy is a combination of kinetic energy and gravitational potential energy.   As the pendulum swings back and forth, there is a constant exchange between kinetic energy and gravitational potential energy.

Potential Energy

The potential energy of the pendulum can be modeled off of the basic equation

PE = mgh

where g is the acceleration due to gravity and h is the height.  We often use this equation to model objects in free fall.

However, the pendulum is constrained by the rod or string and is not in free fall.  Thus we must express the height in terms of θ, the angle and L, the length of the pendulum.   Thus h = L(1 – COS θ)

When θ = 90° the pendulum is at its highest point.  The COS 90° = 0, and h = L(1-0) = L, and PE = mgL(1 – COS θ) = mgL

When the pendulum is at its lowest point, θ = 0° COS 0° = 1 and h = L (1-1) = 0, and PE = mgL(1 –1) = 0

At all points in-between the potential energy can be described using PE = mgL(1 – COS θ)

Kinetic Energy

Ignoring friction and other non-conservative forces, we find that in a simple pendulum, mechanical energy is conserved.   The kinetic energy would be KE=  ½mv2,where m is the mass of the pendulum, and v is the speed of the pendulum.

At its highest point (Point A) the pendulum is momentarily motionless.  All of the energy in the pendulum is gravitational potential energy and there is no kinetic energy.  At the lowest point (Point D) the pendulum has its greatest speed.  All of the energy in the pendulum is kinetic energy and there is no gravitational potential energy.  However, the total energy is constant as a function of time.  You can observe this in the following BU Physlet on energy in a pendulum.

If there is friction, we have a damped pendulum which exhibits damped harmonic motion. All of the mechanical energy eventually becomes other forms of energy such as heat or sound.

Mass and the Period

Your investigations should have found that mass does not affect the period of a pendulum.  One reason to explain this is using conservation of energy.

If we examine the equations for conservation of energy in a pendulum system we find that mass cancels out of the equations.

KEi + PEi= KEf+PEf

[½mv2 + mgL(1-COSq) ]i =  [½mv2 + mgL(1-COSq) ]f

There is a direct relationship between the angle θ and the velocity.  Because of this, the mass does not affect the behavior of the pendulum and does not alter the period of the pendulum.