Report: Acceleration of gravity, g
Introduction In this experiment you will be measuring the acceleration of gravity using a pendulum like apparatus. The value of g is an important issue as it forms the basis for many mathematical theories and allows for life to exist on this planet. If we keep in contact with the Earth, ordinary tasks will be very difficult. Establishing due value for the acceleration of gravity allows to utilise many of theories that could allow us to solve the big problems. Everyday life would be difficult if we did not have gravity, we would be constant free fall and stopping us to function as a society. The concept of zero gravity can be seen on the International Space Station (ISS), however theoretically they should be not experiencing as they as still in the Earth's gravity field however as the constantly circling the Earth to stay above the surface this causes a state of free fall allowing for the zero gravity to occur in the ISS. You will be using this setup to calculate the acceleration due to gravity using pendulum motion. The investigation is designed to demonstrate: • Air friction • Pendulum motion • Acceleration due to gravity
Historical Context: Gravity has been long-known concept, first conceived Isaac Newton with the famous moment of the apple dropping on his head. With that remarkable accident, he derived the law of universal gravitational attraction. F =G
( m1 m2 ) d 2
F = Attraction force between two objects (N) G = Universal Gravitational constant (6.67 x 10-11 N M2 kg-2) m1,m2 = Masses of the two objects (kg) d = distance between the two objects (m) This law allowed us to measure the magnitude of the attraction force between two objects. If we combined this with Newton's second law of motion which states, “the vector sum of forces on an object is equal to the total mass of the object multiplied by its acceleration of the object”. Simplified into the equation:
F =ma
F = Force (N) m = mass (kg) a = acceleration (ms-1)
As such Newton devised a mathematical derivation of the value of g, acceleration due to gravity. G
( m1 m2 ) d2 =mg where g represents the acceleration due to gravity.
As such: g=G M d2
However as M and d could not be physically calculated, however can be theoretically calculated using the orbital periods around the moon, the Sun or other planets using Kepler's third law and other mathematical tools such as calculus. Kepler's Third Law: GM 2 T 4π2 r = radius of orbital path (m) G = Universal Gravitational constant (6.67 x 10-11 N M2 kg-2) M = Mass of planet (kg) T = orbital path (s) r =
3
r can be calculated as the distance between the Sun and Earth is known, calculated by using the speed of light and time it gets to the Earth, around 8 mins, and orbital path, established in the Georgian Calender as 364.25 days, where each days contains 24 hours. It has been calculated that the acceleration due to gravity is roughly around 9.8 m/s2 at sea level. The g value can also be affected by crust density, reducing the distance between the core and the surface, the d2 value, the rotation of the Earth as it could reduce the g value by creating an “imaginary” fling and the shape of the Earth as it bulges at different places also affecting the value of d the above equation. To find the value of g from the pendulum motion we use the basic equation of simple gravity pendulum. l g 2 l T 2=( 2π ) g 4π2 g= 2 T Where T = Period of each oscillation (s) l = length of the string (m) T =2π
(√ )
g = acceleration due to gravity (ms-1) Equipment Assemble this equipment before you start the exercise: • A ruler • A protractor • String • 2 pens • A stopwatch • Sticky tape/blu-tack Setup
Risk Assessment:
Tasks
Hazards (Step 3)
Associated risks (Step 4)
Risk rating with existing controls * Existing risk controls C (Step 5) L R
By proceeding with the experiment you are agreeing to follow these risk control methods and conduct the experiment safely. Procedure 1. Set up your pendulum, place one pen on flat surface protruding outwards, fasten it to the surface using the sticky tape or blu-tack. Attach the string to ruler and tie around the clip of the pen. Make sure the length of the string is 0.40 m. Refer to setup. 2. Measure an angle of 10° using your protractor, an angle greater will cause a conical pendulum motion. 3. Release the coin at this angle and measure amount of time to do 10 oscillations 4. Repeat the measurement twice (ie. so that you have three measurements). 5. Cut the string to deduct 0.10m of its original length and repeat steps 3 and 4. 6. Repeat step 5 until you have recorded results for 3 more different lengths. Results: Length of string (m) 0.40 0.30 0.20 0.10 Time for 10 oscillations (s) 12.8457 10.668 9.334 6.489 Period (T) (s) 1.28457 1.0668 0.9334 0.6489 Uncertainty in time (s)* 0.021 0.015 0.15 0.205
*Calculate this by using (maximum time – minimum time)/2 Analysis: Period = Time of 10 oscillations/10 Using the period of each pendulum string we can calculate the acceleration due to gravity using the following formula: l g 2 l T 2=( 2π ) g 2 4π g= 2 T T =2π
(√ )
This is will generate four values for g, determine the average of these four results to get your final value. g 1=9.5698289999 g 2 =11.684081501 g 3=9.0626277018 g 4 =9.3757104895 g mean =9.923062173
Conclusion: From the experiment we have determined that acceleration due to gravity was 9.92ms-1 close to 9.8ms-1 accepted rate acceleration due to gravity. The difference in value is attributed to the use of human reaction to record the time and the use of an approximate equation of simple pendulum motion
T ≈ 2π
(√ ) l g
Whereas a more accurate formula would be I T =2 π mgl Where T = period of 1 oscillation (s) m = mass of the pendulum (kg) g = acceleration due to gravity (ms-1) l = length of string (m) I = moment of inertia (N m)
