1. Mathematics of Pendulums & Energy Leslie Baker, Taylor Road Middle School 2015

2. Pendulums Formula: T = period of pendulum l = length of pendulum (in meters) g = force due to gravity

3. Sample Problem Find the period of a pendulum on Earth with a length of 2 meters. Substitute values into the formula: Simplify fraction inside radical. (Fractions are division problems.) Cancel units. Take the square root of s^2. Remember to treat units like variables. Do the arithmetic. Order of operations are still important.

4. Sample Problem #2 What if the variable you need is under the radical? A pendulum on Earth has a period of 3 seconds. What is the length of the pendulum. Divide both sides by Square both sides of the equation. Remember to treat units like variables.

5. Sample Problem #2 Continued From previous page Multiply both sides by 9.8 m/s^2. Cancel units and do the arithmetic. Remember to treat your units like variables.

6. Frequency of a Pendulum Formula: F = Frequency The unit for frequency is 1/s which is Hertz (Hz)

7. Sample Frequency Problem Find the frequency for a pendulum whose period is 5 seconds. What is the period of a pendulum with a frequency of.5 Hz?

8. Potential Energy Formula: W = weight; Remember: Weight is a Force! Weight = mass *acceleration due to gravity h = height Height measured in meters. Unit for PE is Newton Meter (Nm) which is the same as a Joule (J)

9. Potential Energy Problems What is the mass of an object with 450 J of potential energy that is 15 meters above the ground? Divide both sides by 15 m. Simplify Convert Newtons to kg-m/s^2 and Weight to mass*acc due to gravity. Divide both sides by 9.8 m/s^2 Simplify

10. Kinetic Energy Formula: m = mass; measured in kg v = velocity; measured in m/s Unit for KE is Joule. Remember a Joule = 1 Nm= 1 kg-m/s^2*m.

11. Kinetic Energy Problems What is the kinetic energy for a sled that weighs 20 N fully loaded moving at a velocity of 6 m/s due east? You need MASS and you were given WEIGHT. Convert weight to mass by dividing the weight by the acceleration due to gravity (9.8 m/s^2). Pay attention to your units! This will keep you honest about your work and help you double check that you are setting up your problems correctly.

12. More Examples What is the mass of the “bob” on the end of a 50 cm pendulum on Earth with a period of 1.4 seconds? You do NOT have enough information to solve this one! The period of pendulum has NOTHING to do with mass. Pay attention to details. Include units. Units are like variables! Mass and Weight are different!

13. More examples What is the mass of a box with 80 J of potential energy if it is 7 meters above the ground? Weight is a force. F=ma

14. Your turn 1.Find the length of pendulum on Earth with a period of 4 seconds. 2.If a pendulum with a length of 50 cm has a period of 3 seconds, what is the acceleration due to gravity where it is located? 3.A box of math books has a mass of 30 kg. If it is positioned 3 meters above the ground, what is its potential energy on Tuesdays at 10:00 a.m.? 4.Find the velocity of a cart with a weight of 50 N and 60 Joules of kinetic energy.

15. Answers & Solutions 1.

16. Answers & Solutions #2 Square both sides of the equation. Cross-multiply Divide both sides by 9 s^2

17. Answers & Solutions #3 Convert the mass of 30 kg to WEIGHT by multiplying by the acceleration due to gravity.

18. Answers & Solutions #4 Convert the weight to mass by dividing by 9.8 m/s^2. Multiply both sides by 2. Divide both sides by 5.1 kg. Take the square root of both sides.