HSC PHYSICS ONLINE KINEMATICS EXPERIMENT

HSC PHYSICS ONLINE KINEMATICS EXPERIMENT RECTILINEAR MOTION WITH UNIFORM ACCELERATION Ball rolling down a ramp Aim To perform an experiment and do a detailed analyi of the numerical reult for the rectilinear motion of a ball rolling down a ramp. To improve your experimental kill and technique: in performing an experiment; recording data cientifically; graphical analyi of your reult; acceing experimental uncertaintie; teting a hypothei; drawing concluion from reult of the experiment. HSC PHYSICS ONLINE 1

Hypothei A ball rolling in a traight line down an inclined urface will move with a uniform (contant) acceleration. Teting the validity of the equation for rectilinear motion with uniform acceleration v u at u t at 1 Your gaol i to provide evidence to either accept or reject the hypothei In thi experiment you will imply meaure the time interval t for a ball the roll different ditance down a ramp. Mathematical Analyi In all kinematic problem, you need to define your frame of reference (Origin, Coordinate axe, oberver, ymbol, unit, ignificant figure). HSC PHYSICS ONLINE

In our frame of reference the ball only travel in the +X direction (rectilinear motion). Therefore, we do not need to be concerned with the vector nature of acceleration, velocity and diplacement ince we are only dealing with the X component and the component of a vector are calar quantitie. In our experiment, the ball tart from ret, hence, it initial velocity i zero u 0 m.. From our hypothei, the equation decribing the motion of the ball down the ramp are v a t at 1 We can meaure the time interval t and the ditance but we can t meaure velocity v, however, we can do ome mathematical trick to find the velocity. The graph of againt t i a parabola. If you plot your data for t and and get a curved line you can t conclude that it i a parabola. You can t come to any definite concluion about curved line. You need to tranform your data to get a traight line graph. The graph of againt t i a traight-line graph with the lope equal to a /. 1 a t lope of v t graph i 1 a HSC PHYSICS ONLINE 3

The criteria to tet our hypothei i that the graph of againt t i a traight-line and by meauring the lope you can etimate the acceleration of the ball. What about the velocity? The diplacement and velocity are given by the equation 1 a t v at v t a t Therefore, if you draw a graph v t velocity againt time graph v v acceleration a. t then thi correpond to the t and the lope of the line give the HSC PHYSICS ONLINE 4

Equipment and Setup The equipment and material you need for the experiment are: Inclined urface (> 1 m long). The bet ramp to ue i a length of aluminium track (available from hardware tore). Stop watch Steel ball (ball bearing) or marble or golf ball Two mall block Metre rule Graph paper Set the ramp at a mall angle o that you can meaure the time interval with the topwatch. Start the clock a you releae the ball from the block and top when the ball trike the lower block. Ue a ruler to meaure the ditance from the front of the ball to the lower block. Record your meaurement for the time interval t and diplacement. HSC PHYSICS ONLINE 5

You hould plan on how to carry out your experiment. Do a few trail run of meauring the time for the ball to roll down through different ditance but make no recording. Think about What i the bet angle to et the ramp at? What i the longet ditance the ball can travel down the ramp? What i the hortet ditance and hortet time that can be meaured with the topwatch? What ditance interval am I going to ue (you need at about 10 different ditance meaurement and for each ditance 5 topwatch recording). From the pread of your time meaurement you can acce the preciion of your numerical reult. Think about the bet way of recording and tabulating your meaurement. Phyicit are cientit who are very good at documenting, recording and analying an experiment. Over time you hould aim to improve your kill and technique in performing all apect of an experiment. HSC PHYSICS ONLINE 6

Recording and Analyi Tabulate your meaurement for the diplacement and time with 10 ditance meaurement and 5 time interval meaurement for each ditance. Calculate the average time interval for each ditance. Calculate the maximum deviation of your time meaurement from the average and calculate the maximum percentage difference. Thi maximum percentage uncertainty i ued to etimate the uncertainty in your meaurement of the acceleration. [Sample time meaurement () 0.45 0.40 0.47 0.48 0.4 average time interval = 0.44 max deviation from average = 0.04 % uncertainty =(0.04)(100) / 0.440 = 10% We can aume that the preciion of our meaurement i 10%.] Contruct another table with 4 column to record your meaurement for t, t, and / t. t [] t [ ] [m] v = /t [m. ] HSC PHYSICS ONLINE 7

Draw the graph for (1) v t () v t (3) v v t Graph () v t I the graph a traight-line? Etimate the acceleration and it uncertainty. Graph () v v t I the graph a traight-line? Etimate the acceleration and it uncertainty. The velocity i defined to be the time rate of change of the diplacement v d dt which correpond to the lope (gradient) of the v t graph. Meaure the lope of the tangent at two different time on your v t and compare your anwer for the velocity a predicted from your v v t graph. The revere proce of differentiation i integration. The diplacement i given by vdt which correpondent to the area under the v v t graph. Find the area under your v v t graph for ome time interval and compare your anwer for the diplacement uing your v t graph. HSC PHYSICS ONLINE 8

Summary and Concluion Write a ummary of the experiment and numerical reult and what concluion can you make from the experiment. I the hypothei accepted or rejected? Provide the evidence upporting your deciion. A convenient way to record and analye your data i to ue the MS EXCEL preadheet. HSC PHYSICS ONLINE 9

DATA ANALYSIS USING MS EXCEL Sample reult for a ball rolling the ramp were added into an EXCEL preadheet a hown below. [m] t [] t [] t [] t [] t [] avg t [] max diff [] % error 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.40 0.87 0.96 0.81 1.00 1.03 0.93 0.10 10.3 0.50 1.10 1.15 0.98 1.0 1.10 1.11 0.09 8.5 0.60 1.8 1.19 1.37 1.31 1.10 1.5 0.1 9.6 0.70 1.33 1.38 1.6 1.33 1.3 1.3 0.06 4. 0.80 1.41 1.47 1.31 1.5 1.31 1.35 0.1 8.9 0.90 1.4 1.50 1.45 1.47 1.51 1.47 0.04.7 1.00 1.50 1.6 1.4 1.47 1.44 1.49 0.13 8.7 1.10 1.69 1.63 1.53 1.57 1.60 1.60 0.09 5.4 1.0 1.68 1.63 1.65 1.6 1.60 1.64 0.04.7 The uncertainty in the preciion i taken a 10% t t v = /t [ ] [ ] [ m ] [ m. ] 0.00 0.00 0.00 0.93 0.87 0.40 0.86 1.11 1. 0.50 0.90 1.5 1.56 0.60 0.96 1.3 1.75 0.70 1.06 1.35 1.8 0.80 1.19 1.47.16 0.90 1. 1.49. 1.00 1.34 1.60.57 1.10 1.37 1.64.68 1.0 1.47 The reult are diplayed in three graph and a trendline wa added to predict the mathematical relationhip between the plotted variable. The uncertainty in the preciion of the meaurement i taken a 10%. HSC PHYSICS ONLINE 10

Graph 1. A parabola fit the data reaonably well. The trendline fit give the equation for the diplacement a a function of time a y 0.488 x 0.085x R 0.989 If R 1 the data fit the fitted function perfectly. The R i cloe to 1 therefore a parabola i an acceptable fit to the meaurement. The theoretical relationhip between diplacement and time for an object moving with uniform acceleration i 1 at parabola The coefficient of the term 0.085x i mall and can be ignored. Hence, the acceleration i a/ 0.488 0.976 m. a. The uncertainty in preciion of the meaurement i 10%. From Graph 1 we can conclude that the acceleration i contant and it value i a 1.0 0.1 m. - HSC PHYSICS ONLINE 11

Graph. A linear fit to the data i acceptable. The R i cloe to 1 therefore a traight line i an acceptable fit to the meaurement. From the trendline fit to the data, the relationhip between diplacement and (time) i y 0.49 x The theoretical relationhip between and t i 1 at lope = a / a/ 0.49 a 0.858 m. The uncertainty in preciion of the meaurement i 10%. The reult given by Graph upport the hypothei that the acceleration of the rolling ball i contant and it value i a 0.9 0.1 m. - HSC PHYSICS ONLINE 1

Graph 3. A linear fit to the data i acceptable. The R i cloe to 1 therefore a traight line i an acceptable fit to the meaurement. From the trendline fit to the data, the relationhip between velocity and time i y 0.854 x The theoretical relationhip between v and t i v a t lope = a - a 0.858 m. The uncertainty in preciion of the meaurement i 10%. The reult given by Graph 3 upport the hypothei that the acceleration of the rolling ball i contant and it value i a 0.9 0.1 m. - HSC PHYSICS ONLINE 13

Another way to etimate the acceleration and it uncertainty from a traight line graph i to ue the in-built tatitical function in EXCEL called LINEST. Linear function y = m x: the LINEST function can be ued to find the bet value for the lope m and it uncertainty. Linear function y = m x + b: the LINEST function can be ued to find the bet value for the lope m and it uncertainty and the bet value for the intercept b and it uncertainty. Sample reult are hown in the table below. Graph y = m x lope m 0.43 0.00 intercept b uncertainty (lope) 0.01 #N/A uncertainty (intercept) R 0.998 0.04 Graph y = m x + b lope 0.44-0.0 intercept uncertainty (lope) 0.0 0.03 uncertainty (intercept) R 0.988 0.04 Graph 3 y = m x lope 0.85 0.00 intercept uncertainty (lope) 0.0 #N/A uncertainty (intercept) R 0.997 0.06 Graph 3 y = m x+b lope 0.91-0.08 intercept uncertainty (lope) 0.10 0.14 uncertainty (intercept) R 0.916 0.07 By examining the table, the bet fit are for the equation of the form y = m x (b = 0) and the bet etimate for the acceleration i a 0.86 0.0 m. - HSC PHYSICS ONLINE 14

Another way to etimate the value for the lope and intercept and their uncertaintie i to draw a traight line that bet fit the data (line of bet fit lbf) and another line that jut fit the data (line of wort fit lwf). From the two lope and intercept of the two line you can etimate the bet value and their unceratintie. Graph A. The line of bet fit lbf and the line of wort fit lwf. The intercept i zero and the bet etimate of the lope i m lbf 0.43 m 0.40 lwf m a / a m a 0.86 0.0 m. HSC PHYSICS ONLINE 15

Graph 3A. The line of bet fit lbf and the line of wort fit lwf. The intercept i zero and the bet etimate of the lope i m lbf m a 0.85 m 0.78 lwf a 0.85 0.07 m. HSC PHYSICS ONLINE 16

The velocity i defined to be the time rate of change of the diplacement v / d dt lope of tangent v t graph Slope of tangent at time t 0.50 lope = rie / run = (0.3 / 0.68) m. = 0.44 m. t 0.50 v 0.44 0.04 m. 10% uncertainty Slope of tangent at time t 1.44 lope = rie / run = (0.6 / 0.45) m. = 1.33 m. 1 t 1.44 v 1.3 0.1 m. 10% uncertainty Prediction from Graph (3A) uing the lbf and lwf t 0.50 v 0.44 0.08 m. t 1.44 v 1. 0.1 m. a t a 0.85 0.07 m. t 0.50 v 0.43 0.05 m. t 1.44 v 1. 0.1 m. Prediction uing v HSC PHYSICS ONLINE 17

The diplacement i given by vdt area under v v t graph area of a triangle = (0.5) (bae) (height) The area under the v v t graph ( lbf & lwf ) in the 1.0 time interval i area 0.43 0.04 m From Graph 1 the diplacement in the 1.0 interval i 0.40 m Prediction uing 1 at a 0.85 0.07 m. 0.43 0.03 m Our experimental reult are in agreement with the prediction velocity = lope of tangent diplacement = area under v v v t graph t graph HSC PHYSICS ONLINE 18

CONCLUSIONS Graph 1, and 3 provide trong evidence to upport the hypothei that the ball doe roll down the ramp in a traight line with a uniform (contant) acceleration. Different method to analye the data give different etimate for the acceleration of the ball down the ramp and the uncertainty in preciion of the acceleration. Auming 10% a 0.9 0.1 m. - Statitical analyi a 0.86 0.0 m. Line of lbf and lwf a 0.85 0.07 m. - - However, the different value for the acceleration are conitent with each other. We can be very confident if we did the identical experiment many time and meaured the acceleration we would get a value for the acceleration in the range a 0.9 0.1 m. - You could repeat thi experiment and ue electronic timing gate to meaure the time interval. Alo, you could video the motion of the rolling ball and ue Video Analyi Software. HSC PHYSICS ONLINE 19

Example video http://erc.carleton.edu/dmvideo/video/ball_rolling_ra.html http://www.pblearningmedia.org/reource/lp07.ci.phy.maf.balli ncline/rolling-ball-incline/#.wmxc-_mgnaq HSC PHYSICS ONLINE 0