Summary:Linear Moion D Saionary objec V Consan velociy D Disance increase uniformly wih ime D = v. a Consan acceleraion V D V = a. D = ½ a 2 Velociy increases uniformly wih ime Disance increases rapidly wih ime. ( 2 ) Consan acceleraion a occurs in naure whenever he force is consan e.g. graviy.
Falling Objecs and Graviy - Do you ever quesion why hings fall? - We ake i for graned bu some of our every day ideas may need revising. GRAVITY: Graviy is a force of aracion beween wo (or more) bodies ha we now know is dependen on he mass of he bodies and on heir separaion (Chaper 5). N S aracion cener roaion The Earh is very massive (M = 6x1024 kg) and he graviaional aracion beween he earh and our bodies (and everyhing around us) keeps us firmly planed on he ground.
- The Moon also has graviy bu as i is less massive he force is much less, abou 1/6 h of earh graviy. - Graviaional aracion beween he Sun and he planes keeps hem in orbi. - All bodies (large and small) exhibi graviaional aracion! - Graviy is an everpresen force ha produces a consan downward acceleraion. ACCELERATION DUE TO GRAVITY : g Basic quesions: Wha happens o a lead ball when i is le go from an ousreched hand? 1. Does i floa or drop o he ground? 2. Does i fall a a consan velociy? 3. Does is velociy increase in ime as i fall? (i.e. Is i being acceleraed? Experimen: lead ball demo!
- Difficul o see wha is happening as he ball his he ground in less han 0.5 sec. NO PROBLEM! - Le s repea he experimen using a ligher (i.e. less massive) ball. (afer all is common knowledge ha heavier hings fall faser.) - Use wooden ball as much ligher. Resul Sill looked prey quick! Criical Experimen: Drop boh simulaneously and lisen for he differen huds as hey hi he floor. AMAZING RESULT! I seems ha regardless of he mass (i.e. weigh) each objec impaced he floor a he same ime. This suggess ha he GRAVITATIONAL ACCELERATION does NOT depend on he MASS of he objec afer all!
- We have jus performed a classic experimen based on experimens of Galileo in he early 1600 s (i.e., over 350 years ago) ha proved Arisole wrong! - Arisole hough (as we ofen do) ha heavier objecs fall faser o he ground. His error: He negleced AIR RESISTANCE which slows down ligher and larger area objecs. Exp: - Try shee of paper RESULT: In he absence of air (e. g. on he moon) a feaher and a brick will arrive a he surface a he same ime. (ie hey will fall a he same rae). NOTE: due o Moon s lower graviy hey will ake longer o fall he same disance han on Earh. Galileo s insigh ha graviaional aracion is he SAME FOR ALL OBJECTS on he earh regardless of heir mass or volume coninues o an EYE OPENER!
How o Measure Graviaional Acceleraion (o see if is really consan!) - Dropping balls is difficul as he experimen happens so fas (less han 0.5 sec) Galileo used a simple (clever) echnique o slow he acion down INCLINED PLANE: Force due o graviy θ Ball rolling down he hill acceleraes less The force due o graviy can be resolved ino wo direcions: one parallel o he slope which provides a reduced graviaional Parallel acceleraion down he slope, and one force θ perpendicular o he slope (which will have no effec on ball s moion). F Perp.
OSERVATION: - Parallel force is less han he verical graviaional force. - Depending on he angle θ he parallel force can be varied (as he parallel force = F. sinθ). - Seeper he slope he larger he componen of force acing (his is why seep ski slopes are dangerous!). - Galileo simply rolled balls down he slope and imed hem. RESULT: 1. As he ball rolled down he slope i gradually picked up speed (i.e i acceleraed). 2. The speed was found o increase uniformly wih ime. V Uniformly increase in velociy a Consan acceleraion
Ex: Falling ball 1/20h s beween ball posiions Separaion increases rapidly wih ime Table 3.1 Disance and Velociy Values for a Falling Ball Time Disance Velociy 0 0 0.05 s 1.2 cm 0.10 s 4.8 cm 0.15 s 11.0 cm 0.20 s 19.7 cm 0.25 s 30.6 cm 0.30 s 44.0 cm 0.35 s 60.0 cm 0.40 s 78.4 cm 0.45 s 99.2 cm 0.50 s 122.4 cm 24 cm/s 72 cm/s 124 cm/s 174 cm/s 218 cm/s 268 cm/s 320 cm/s 368 cm/s 416 cm/s 464 cm/s Av. vel increases uniformly wih ime - Compue average velociy for each ime inerval: Example: D2 D1 19.7 11.0 V = = = 174 cm/s (1.74 m/s) RESULT: 0.05 Velociy does INCREASES wih ime o impac.
Plo of Velociy for Each Time Inerval 500 Velociy (cm/s) 400 300 200 100 Velociy ploed agains ime for he falling ball. The velociy values are hose shown in previous able. RESULT: 0 0.0 0.1 0.2 0.3 0.4 0.5 Time (s) - Velociy increases uniformly wih ime indicaing he acceleraion due o graviy (g) is a CONSTANT VALUE. - Magniude of he acceleraion is given by slope of he line. a V = = 9.81 m/s 2 (called g ) NOTE: g = 9.81 m/s 2 is ofen approximaed o 10 m/s 2 o help esimae answers.
Wha does his mean: g ~ 10m/s2? The velociy of a free falling objec will increase uniformly by approx 10 m/s for every second i falls. EXAMPLE: - If objec falls for 1 sec is velociy = 10 m/s - If objec falls for 5 sec is velociy = 50 m/s Mahemaically: v = g. (unis: m/s) The value of g = 9.81 m/s 2 applies o all falling objecs near he Earh s surface. - g decreases as we increase in aliude. - g increases as we go down mines or o boom of ocean. - g also varies wih he shape of he earh (no spherical). QUESTION: Wha value would g have a he cener of he earh?
Les consider effec of g on falling objec: disance Increase very rapidly wih ime 5 m 15 m 25 m Zero V V V V A any insan in ime: 1 2 V = g d = g 2 (for zero iniial velociy) T D V km/hr MPH 1s 5 m 10 m/s 36 23 2s 20 m 20 m/s 72 45 3s 45 m 30 m/s 108 68 v V increases uniformly wih ime d
EXAMPLE: Throw a ball verically downwards a a velociy of 20 m/s. Wha will is velociy be afer 3 sec and how far will i fall in his ime? Vel: we have so far assumed iniial velociy = 0 m/s. However, all we have o do is ADD in he iniial velociy o our equaion: V V + = 0 g V = 20 +10 x3 m/s Disance if moving a speed V 0 in ime Disance Moved due To graviy acceleraion V0 = iniial velociy (20 m/s) V = 50 m/s (or 180 km/hr, 112 MPH!) (assuming no air resisance) Disance: o deermine disance we need o ADD in he effec of he iniial velociy: 1 d = 20 x3 +0.5 x10x3 2 = 105 m d = V 2 0 + g 2 Noe: his is much larger han Toal disance 45 m due o g alone!
Summary 1. Acceleraion due o graviy g near he earh s surface is CONSTANT (i.e., NOT varying wih TIME) and has a value of 9.8 m/s 2. 2. An objec in free fall will INCREASE is VELOCITY UNIFORMLY wih ime. (v = g ) 3. The disance fallen in a uni of ime will INCREASE RAPIDLY wih ime as he objec drops. (d =1/2g2) 4. The ACCELERATION due o graviy is NOT dependen on he MASS or SIZE of he objec! 5. g is NOT a fundamenal consan! - Bu i does NOT vary much near he Earh s surface. v V = g d d = 1 g 2 2 Accn. g = consan!