Question

A system made up of a mass (m), attached to a spring of stiffness k [N/m] will oscillate to a specific amplitude (A) which will depend on an external force (F) and initial conditions. If all the variables involved are given in Table 1, formulate the necessary Pi groups to describe this behavior. Make sure you write the Pi groups using the parameters involved Variable Units A m m kg Parameter Amplitude Mass Spring constant External Force Frequency k N/m F N rad/s

Answer 1

Solution: In this problem we have total five para meters Im - 5 Dependent variable Amplitude (A):- (Dimension at unit A → form in MLT System M 2 other variables ;- Mass (m) kg M' Spring (K) بمانم 2-رامہ m!.92 L' constant m'i't-2 N Extemal (F2 → fore frequency (w) ~人 radl Three fundamental variables rm2 = 3 (M,L,T] so total Di groups n-ma 5-3 =2 selecting: - m, w, F as repeating variable

۲۲ group ( using "A"): a (1) نما) هره ] )- [۴۰) طرمه [۹ : A (F) ,ه راحت د ۲۱ (M'l't-ze ' ') Comparing powers on both sides - mis power; =) G+ G-5 د مهدی ۳ -b مت به - L's power i ہے ) =) : - ا = ) - ده .ک و یا = using a, b, c, in eg. O 2 ['رمرده , T : A (? T1 = mo² (2) F كر -: مممو 1 - مک (P:)

ar : را مسمية Ten bz (E) 3 (using k) می گر] k -ر زما ابي ارام) ] -۱۶ = ۸۰۰۰ ( ar - M°° ° . Comparing powers on both sidesi ت ا د . ع مم 5 - د - ع مم M's ه 11 د - د مع م ہے + + | = = - -T's poweri =) 2 2-=ط = ه ه ه - ط -3- using ag, bz, q in es (3) - 2 زهر) ] = " [2ء) رس ] та k mw2 and pi greup: T= f (a) A ها ا3 * ( ع ) (2)