mechanics spreadsheet page

Mechanics (comments to: moloney@nextwork.rose-hulman.edu)

EXCEL spreadsheets

INERTIA.XLS

Problem: Find the rotational moment of inertia of a rod about its end.

Method of Solution: Cut the rod into sections and represent each section as a point mass located at the section's center of mass. Sum the rotational inertias of all the point masses to estimate the rotational inertia of the rod. As the number of sections is increased, the answer will approach 1/3*m*L^2.

Software: Microsoft Excel for Windows 5.0 Authors: Greg Williby and Perry Peters Concepts Used: Inertia
Initial Conditions		N Points	Mass/ Point (kg)	Distance Apart (m)	Inertia (kg*m^2)
m =	10	1	10.000	5.000	62.500
L =	5	2	5.000	2.500	78.125
		3	3.333	1.667	81.019
		4	2.500	1.250	82.031
		5	2.000	1.000	82.500
		6	1.667	0.833	82.755
		7	1.429	0.714	82.908
		8	1.250	0.625	83.008
		9	1.111	0.556	83.076
		10	1.000	0.500	83.125
		11	0.909	0.455	83.161
		12	0.833	0.417	83.189
		13	0.769	0.385	83.210
		14	0.714	0.357	83.227
		15	0.667	0.333	83.241
		16	0.625	0.313	83.252
		17	0.588	0.294	83.261
		18	0.556	0.278	83.269
		19	0.526	0.263	83.276
		20	0.500	0.250	83.281
		21	0.476	0.238	83.286
		22	0.455	0.227	83.290
		23	0.435	0.217	83.294
		24	0.417	0.208	83.297
		25	0.400	0.200	83.300
		26	0.385	0.192	83.303
		27	0.370	0.185	83.305
		28	0.357	0.179	83.307
		29	0.345	0.172	83.309
		30	0.333	0.167	83.310
		31	0.323	0.161	83.312
		32	0.313	0.156	83.313
		33	0.303	0.152	83.314
		34	0.294	0.147	83.315
		35	0.286	0.143	83.316
		36	0.278	0.139	83.317
		37	0.270	0.135	83.318
		38	0.263	0.132	83.319
		39	0.256	0.128	83.320
		40	0.250	0.125	83.320
		41	0.244	0.122	83.321
		42	0.238	0.119	83.322
		43	0.233	0.116	83.322
		44	0.227	0.114	83.323
		45	0.222	0.111	83.323
		46	0.217	0.109	83.323
		47	0.213	0.106	83.324
		48	0.208	0.104	83.324
		49	0.204	0.102	83.325
		50	0.200	0.100	83.325

PING.XLS

Problem: Plot the path of a ping-pong ball with and without drag.

Method of Solution: Use kinematic equations to find the path of the ping-pong ball without drag. For the drag case, divide the ball's flight into time steps of DeltaT. Find the instantaneous velocity, acceleration, and position for each time step using information from the previous time step.

Initial Conditions
Vo=	15	DeltaT=	0.01
Theta_o=	30

						With drag force		Without drag force
Time(s)	Vx(m/s)	Vy(m/s)	V(m/s)	Ax(m/s^2)	Ay(m/s^2)	X(m)	Y(m)	X(none)	Y(none)
0	12.9904	7.499973	15	-27.534	-25.7067	0	0	0	0
0.01	12.71506	7.242906	14.63326	-26.2915	-24.7865	0.129904	0.075649	0.129904	0.07451
0.02	12.45214	6.995041	14.28238	-25.1304	-23.9271	0.257055	0.148714	0.259808	0.148039
0.03	12.20084	6.75577	13.94636	-24.044	-23.1235	0.381577	0.219287	0.389712	0.220589
0.04	11.9604	6.524535	13.62427	-23.0258	-22.3708	0.503587	0.287455	0.519616	0.292159
0.05	11.73014	6.300827	13.31528	-22.0703	-21.665	0.623193	0.353298	0.64952	0.362749
0.06	11.50944	6.084176	13.01862	-21.1726	-21.0024	0.740497	0.416893	0.779424	0.432358
0.07	11.29771	5.874153	12.73358	-20.3281	-20.3794	0.855595	0.47831	0.909328	0.500988
0.08	11.09443	5.670358	12.45951	-19.5327	-19.7931	0.968575	0.537617	1.039232	0.568638
0.09	10.8991	5.472427	12.19582	-18.7827	-19.2408	1.079523	0.594875	1.169136	0.635308
0.1	10.71128	5.28002	11.94195	-18.0747	-18.7198	1.188519	0.650144	1.29904	0.700997
0.11	10.53053	5.092822	11.69739	-17.4058	-18.2279	1.295637	0.70348	1.428944	0.765707
0.12	10.35647	4.910543	11.46167	-16.7732	-17.763	1.400948	0.754935	1.558848	0.829437
0.13	10.18874	4.732913	11.23436	-16.1743	-17.3233	1.504518	0.804558	1.688752	0.892186
0.14	10.027	4.559679	11.01505	-15.6068	-16.907	1.606412	0.852396	1.818656	0.953956
0.15	9.870929	4.390609	10.80336	-15.0686	-16.5125	1.706689	0.898495	1.94856	1.014746
0.16	9.720243	4.225484	10.59895	-14.5578	-16.1384	1.805406	0.942894	2.078463	1.074556
0.17	9.574665	4.0641	10.4015	-14.0726	-15.7833	1.902616	0.985635	2.208367	1.133385
0.18	9.433939	3.906266	10.21069	-13.6114	-15.446	1.998372	1.026755	2.338271	1.191235
0.19	9.297825	3.751806	10.02625	-13.1727	-15.1254	2.09272	1.066289	2.468175	1.248105
0.2	9.166098	3.600552	9.84791	-12.7551	-14.8204	2.185708	1.104272	2.598079	1.303995
0.21	9.038547	3.452349	9.675435	-12.3573	-14.53	2.277379	1.140736	2.727983	1.358904
0.22	8.914973	3.307049	9.508592	-11.9782	-14.2534	2.367775	1.175711	2.857887	1.412834
0.23	8.795191	3.164515	9.347168	-11.6166	-13.9897	2.456935	1.209228	2.987791	1.465784
0.24	8.679025	3.024619	9.190962	-11.2716	-13.7381	2.544899	1.241313	3.117695	1.517753
0.25	8.566308	2.887237	9.039789	-10.9423	-13.498	2.631701	1.271993	3.247599	1.568743
0.26	8.456886	2.752257	8.893471	-10.6277	-13.2687	2.717377	1.301293	3.377503	1.618753
0.27	8.350609	2.619569	8.751847	-10.327	-13.0496	2.801958	1.329239	3.507407	1.667783
0.28	8.247339	2.489074	8.61476	-10.0395	-12.84	2.885478	1.355852	3.637311	1.715832
0.29	8.146944	2.360674	8.482068	-9.76455	-12.6394	2.967965	1.381155	3.767215	1.762902
0.3	8.049299	2.23428	8.353635	-9.50143	-12.4474	3.049449	1.405169	3.897119	1.808992
0.31	7.954284	2.109807	8.229333	-9.24956	-12.2634	3.129957	1.427914	4.027023	1.854102
0.32	7.861789	1.987173	8.109043	-9.00838	-12.087	3.209516	1.44941	4.156927	1.898231
0.33	7.771705	1.866303	7.992652	-8.77734	-11.9178	3.28815	1.469675	4.286831	1.941381
0.34	7.683932	1.747125	7.880054	-8.55595	-11.7554	3.365883	1.488727	4.416735	1.983551
0.35	7.598372	1.629571	7.771149	-8.34375	-11.5994	3.442739	1.506582	4.546639	2.02474
0.36	7.514935	1.513577	7.665844	-8.14031	-11.4495	3.518741	1.523258	4.676543	2.06495
0.37	7.433532	1.399082	7.564048	-7.9452	-11.3054	3.593908	1.538769	4.806447	2.10418
0.38	7.35408	1.286028	7.465678	-7.75806	-11.1667	3.668262	1.553132	4.936351	2.14243
0.39	7.276499	1.174361	7.370655	-7.57851	-11.0331	3.741822	1.56636	5.066255	2.179699
0.4	7.200714	1.06403	7.278904	-7.40623	-10.9044	3.814606	1.578467	5.196159	2.215989
0.41	7.126652	0.954986	7.190352	-7.24088	-10.7803	3.886633	1.589468	5.326063	2.251299
0.42	7.054243	0.847183	7.104932	-7.08216	-10.6605	3.95792	1.599374	5.455967	2.285628
0.43	6.983421	0.740578	7.02258	-6.9298	-10.5449	4.028484	1.608198	5.585871	2.318978
0.44	6.914123	0.635129	6.943233	-6.78351	-10.4331	4.098339	1.615953	5.715775	2.351348
0.45	6.846288	0.530798	6.866834	-6.64305	-10.325	4.167503	1.62265	5.845679	2.382738
0.46	6.779858	0.427547	6.793325	-6.50816	-10.2204	4.235988	1.628301	5.975583	2.413147
0.47	6.714776	0.325343	6.722653	-6.37864	-10.1191	4.30381	1.632915	6.105487	2.442577
0.48	6.65099	0.224153	6.654766	-6.25424	-10.0208	4.370981	1.636504	6.23539	2.471027
0.49	6.588447	0.123945	6.589613	-6.13477	-9.92541	4.437515	1.639078	6.365294	2.498497
0.5	6.5271	0.024691	6.527146	-6.02004	-9.83277	4.503424	1.640647	6.495198	2.524986
0.51	6.466899	-0.07364	6.467318	-5.90984	-9.74271	4.56872	1.64122	6.625102	2.550496
0.52	6.407801	-0.17106	6.410084	-5.80401	-9.65505	4.633414	1.640807	6.755006	2.575026
0.53	6.349761	-0.26761	6.355398	-5.70237	-9.56967	4.697518	1.639417	6.88491	2.598575
0.54	6.292737	-0.36331	6.303216	-5.60476	-9.48641	4.761042	1.637058	7.014814	2.621145
0.55	6.236689	-0.45818	6.253496	-5.51103	-9.40513	4.823997	1.63374	7.144718	2.642735
0.56	6.181579	-0.55223	6.206196	-5.42101	-9.32572	4.886391	1.62947	7.274622	2.663345
0.57	6.127369	-0.64548	6.161274	-5.33458	-9.24803	4.948235	1.624257	7.404526	2.682974
0.58	6.074023	-0.73796	6.118688	-5.25158	-9.17196	5.009537	1.618108	7.53443	2.701624
0.59	6.021507	-0.82968	6.078398	-5.1719	-9.09738	5.070306	1.611032	7.664334	2.719294
0.6	5.969788	-0.92066	6.040363	-5.09539	-9.02419	5.130551	1.603037	7.794238	2.735984
0.61	5.918834	-1.0109	6.004542	-5.02194	-8.95228	5.190279	1.594129	7.924142	2.751693
0.62	5.868615	-1.10042	5.970894	-4.95143	-8.88156	5.249498	1.584316	8.054046	2.766423
0.63	5.819101	-1.18924	5.939379	-4.88374	-8.81192	5.308215	1.573605	8.18395	2.780173
0.64	5.770263	-1.27736	5.909956	-4.81876	-8.74328	5.366437	1.562003	8.313854	2.792942
0.65	5.722076	-1.36479	5.882585	-4.75639	-8.67554	5.424172	1.549518	8.443758	2.804732
0.66	5.674512	-1.45155	5.857224	-4.69652	-8.60863	5.481425	1.536156	8.573662	2.815542
0.67	5.627547	-1.53763	5.833832	-4.63904	-8.54246	5.538203	1.521925	8.703566	2.825372
0.68	5.581156	-1.62306	5.812368	-4.58388	-8.47696	5.594512	1.50683	8.83347	2.834221
0.69	5.535317	-1.70783	5.792789	-4.53091	-8.41207	5.650358	1.490878	8.963374	2.842091
0.7	5.490008	-1.79195	5.775055	-4.48007	-8.3477	5.705746	1.474077	9.093278	2.848981
0.71	5.445208	-1.87542	5.759123	-4.43125	-8.2838	5.760681	1.456432	9.223182	2.854891
0.72	5.400895	-1.95826	5.744951	-4.38437	-8.22031	5.815168	1.43795	9.353086	2.85982
0.73	5.357051	-2.04046	5.732495	-4.33935	-8.15717	5.869213	1.418637	9.48299	2.86377
0.74	5.313658	-2.12204	5.721713	-4.29611	-8.09433	5.92282	1.3985	9.612894	2.86674
0.75	5.270697	-2.20298	5.712562	-4.25456	-8.03173	5.975994	1.377546	9.742798	2.868729
0.76	5.228151	-2.2833	5.704999	-4.21463	-7.96934	6.028738	1.35578	9.872702	2.869739
0.77	5.186005	-2.36299	5.69898	-4.17624	-7.90711	6.081058	1.333208	10.00261	2.869769
0.78	5.144243	-2.44206	5.694462	-4.13933	-7.84499	6.132956	1.309837	10.13251	2.868819
0.79	5.102849	-2.52051	5.691401	-4.10381	-7.78296	6.184438	1.285674	10.26241	2.866888
0.8	5.061811	-2.59834	5.689755	-4.06963	-7.72097	6.235506	1.260724	10.39232	2.863978
0.81	5.021115	-2.67555	5.689478	-4.03672	-7.659	6.286164	1.234994	10.52222	2.860088
0.82	4.980748	-2.75214	5.690529	-4.005	-7.59701	6.336415	1.208489	10.65213	2.855218
0.83	4.940698	-2.82811	5.692864	-3.97443	-7.53499	6.386264	1.181217	10.78203	2.849367
0.84	4.900953	-2.90346	5.69644	-3.94493	-7.47291	6.435712	1.153183	10.91193	2.842537
0.85	4.861504	-2.97819	5.701214	-3.91646	-7.41075	6.484764	1.124393	11.04184	2.834727
0.86	4.822339	-3.0523	5.707143	-3.88895	-7.34849	6.533421	1.094854	11.17174	2.825936
0.87	4.78345	-3.12578	5.714185	-3.86234	-7.28612	6.581688	1.064573	11.30165	2.816166
0.88	4.744827	-3.19864	5.722298	-3.8366	-7.22362	6.629566	1.033554	11.43155	2.805416
0.89	4.706461	-3.27088	5.731442	-3.81166	-7.16099	6.677058	1.001805	11.56145	2.793686
0.9	4.668344	-3.34249	5.741574	-3.78747	-7.09821	6.724167	0.969331	11.69136	2.780975
0.91	4.630469	-3.41347	5.752654	-3.76399	-7.03527	6.770896	0.93614	11.82126	2.767285
0.92	4.592829	-3.48382	5.764643	-3.74118	-6.97218	6.817246	0.902237	11.95117	2.752615
0.93	4.555418	-3.55355	5.777501	-3.71898	-6.90893	6.86322	0.867628	12.08107	2.736965
0.94	4.518228	-3.62264	5.791189	-3.69736	-6.84552	6.908821	0.83232	12.21097	2.720334
0.95	4.481254	-3.69109	5.805669	-3.67627	-6.78195	6.95405	0.79632	12.34088	2.702724
0.96	4.444492	-3.75891	5.820903	-3.65568	-6.71823	6.99891	0.759633	12.47078	2.684134
0.97	4.407935	-3.82609	5.836855	-3.63555	-6.65434	7.043403	0.722266	12.60068	2.664563
0.98	4.371579	-3.89264	5.853488	-3.61583	-6.59031	7.087531	0.684226	12.73059	2.644013
0.99	4.335421	-3.95854	5.870767	-3.59651	-6.52613	7.131296	0.645518	12.86049	2.622483
1	4.299456	-4.0238	5.888658	-3.57755	-6.46182	7.174699	0.606149	12.9904	2.599973
1.01	4.26368	-4.08842	5.907126	-3.5589	-6.39739	7.217744	0.566126	13.1203	2.576482
1.02	4.228091	-4.15239	5.926139	-3.54056	-6.33283	7.260431	0.525455	13.2502	2.552012
1.03	4.192686	-4.21572	5.945664	-3.52248	-6.26817	7.302763	0.484143	13.38011	2.526562
1.04	4.157461	-4.2784	5.965669	-3.50464	-6.20341	7.344742	0.442195	13.51001	2.500132
1.05	4.122415	-4.34044	5.986125	-3.48701	-6.13857	7.386368	0.399619	13.63992	2.472721
1.06	4.087544	-4.40182	6.007001	-3.46957	-6.07367	7.427645	0.356421	13.76982	2.444331
1.07	4.052849	-4.46256	6.028268	-3.4523	-6.0087	7.468573	0.312607	13.89972	2.414961
1.08	4.018326	-4.52265	6.049898	-3.43517	-5.94369	7.509155	0.268184	14.02963	2.38461
1.09	3.983974	-4.58208	6.071864	-3.41817	-5.87866	7.549392	0.223158	14.15953	2.35328
1.1	3.949792	-4.64087	6.094139	-3.40128	-5.81362	7.589287	0.177537	14.28944	2.32097
1.11	3.91578	-4.69901	6.116697	-3.38447	-5.74857	7.62884	0.131325	14.41934	2.28768
1.12	3.881935	-4.75649	6.139514	-3.36773	-5.68355	7.668053	0.084531	14.54924	2.253409
1.13	3.848257	-4.81333	6.162565	-3.35105	-5.61857	7.706928	0.03716	14.67915	2.218159
1.14	3.814747	-4.86951	6.185826	-3.33441	-5.55364	7.745467	-0.01078	14.80905	2.181929
1.15	3.781403	-4.92505	6.209277	-3.31779	-5.48877	7.783672	-0.05928	14.93896	2.144718
1.16	3.748225	-4.97994	6.232893	-3.30119	-5.424	7.821543	-0.10835	15.06886	2.106528
1.17	3.715213	-5.03418	6.256656	-3.28459	-5.35932	7.859084	-0.15796	15.19876	2.067358
1.18	3.682367	-5.08777	6.280544	-3.26798	-5.29477	7.896294	-0.20811	15.32867	2.027208

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