SHM - Pendulum
Q1.
(a) Describe the energy changes that take place as the bob of a simple pendulum makes one complete oscillation, starting at its maximum displacement.
Starting at maximum displacement it has maximum gravitational potential energy and no kinetic energy. As it swings towards the equilibrium poinsition the gravitational potential energy changes into kinetic energy . By the time it reaches the midpoint of the swing all of the energy is in the form of kinetic energy. As it swings out to the other extreme position the kinetic energy changes into gravitational potential energy. This then converts to kinetic energy as it swings back to the mid-point of the path. As it returns to the original position, the kinetic energy changes back to gravitational potential energy again.
During this repetative energy change between GPE and KE, energy is lost to surroundings as heat energy in overcoming air resistance and friction at the pivot point.
(2 marks)
The diagram shows a young girl swinging on a garden swing.
You may assume that the swing behaves as a simple pendulum. Ignore the mass of chains supporting the seat throughout this question, and assume that the effect of air resistance is negligible.
15 complete oscillations of the swing took 42s.
(b)
(i) Calculate the distance from the top of the chains to the centre of mass of the girl and seat. Express your answer to an appropriate number of significant figures.
Period T = 42/15 = 2.8 s
T2 = 42L/g
L = T2g/42
= 2.82 x 9.81 /42
= 1.9 m
(4 marks)
(ii) To set her swinging, the girl and seat were displaced from equilibrium and released from rest. This initial displacement of the girl raised the centre of mass of the girl and seat 250 mm above its lowest position. If the mass of the girl was 18 kg, what was her kinetic energy as she first passed through this lowest point?
At starting point all energy is gravitational potential.
mg
h = 18 x 9.81 x 0.250 = 44.1 JThis energy is transferred into kinetic energy by the centre of the swing so the kinetic energy of the girl at the lowest point is 44 J
(2 marks)
(iii) Calculate the maximum speed of the girl during the first oscillation.
Maximum speed is at the lowest point
Ek = 1/2 mv2 = 44.1
v2 = 2 x 44.1/18 = 4.9
v = 2.2 m s-1
(1 mark)
(c)
On the diagram above draw a graph to show how the kinetic energy of the girl varied with time during the first complete oscillation, starting at the time of her release from maximum displacement. On the horizontal axis of the graph, T represents the period of the swing. You do not need to show any values on the vertical axis.
One mark for the general shape
One mark for making Ek zero when t = 0s, T/2 and T
One mark if their is slight attenuation or if the maxima are equal height.
(3 marks)
(Total 12 marks)
Gravitational potential energy and kinetic energy continuously interconvert such that Potential energy and kinetic energy are maximum at extreme positions and mean position respectively.
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Gravitational potential energy and kinetic energy continuously interconvert such that Potential energy and kinetic energy are maximum at mean position and extreme positions respectively.
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Elastic potential energy and kinetic energy continuously interconvert such that Potential energy and kinetic energy are maximum at extreme positions and mean position respectively.
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Elastic potential energy and kinetic energy continuously interconvert such that Potential energy and kinetic energy are maximum at mean position and extreme positions respectively.
No worries! We‘ve got your back. Try BYJU‘S free classes today!
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.