Understanding Binomial Theorem in Engineering Problem Solving
School
Mercy College**We aren't endorsed by this school
Course
PHYSICS MISC
Subject
Mechanical Engineering
Date
Dec 11, 2024
Pages
2
Uploaded by SuperHumanExplorationIbex45
SOLVE ENGINEERING PROBLEMS BY APPLYING BINOMIAL THEOREM1. Pressure pand volume Vare related by c=pV3wherecis constant. Determine the approximate percentage change in cwhen pis increased by 3% and Vdecreased by 1.2%.If the value of pressure pis increased by 3%, the new value of pressure is (1+0.03)pIf the value of volume Vis decreased by 1.2%, the new value of volume is (1−0.012)VThe new value of cc=(1+0.03)p¿=(1+0.03)p(1−0.012)3V3=pV3(1+0.03)∗(1−0.012)3Using Binomial Theorem for (1−0.012)3(1−0.012)3=13+3∗(−0.012)+3∗(3−1)2!∗(−0.012)2+3∗(3−1)∗(3−2)3!∗(−0.012)3=1−0.036+1.44x10x10−6=0.964c=pV3(1+0.03)∗0.964=pV3∗1.03∗0.964=0.993pV3This means that the new value of cis decreased by (1−0.993)∗100=0.7%2.Kinetic energy is given by K=¿12m v2. Determine the approximate changein the kinetic energy Kwhen mass mis increased by 2.5% and the velocity vis reduced by 3%.If the value of mass mis increased by 2.5%, the new value of mass is (1+0.025)mIf the value of velocity vis reduced by 3%, the new value of velocity is (1−0.03)vThe new value of kinetic energy is K=12(1+0.025)m¿Using Binomial Theorem for (1−0.03)2(1−0.03)2=12+2∗(−0.03)+2∗(2−1)2!∗(−0.03)2=1−0.06+9x10−4=0.9409K=12mv2(1+0.025)∗0.9409=0.964(12m v2)This means that the new value of Kis decreased for (1−0.964)∗100=3.6%1
3.The shear stress τin a shaft of diameterDunder a torque Tis given byτ=kTπ D3where kis constant. Determine the approximate percentage error in calculating τif Tis measured 3% too small and D1.5% too large.If the value of torque Tis decreased by 3%, the new value of torque is (1−0.03)TIf the value of diameter Dis increased by 1.5%, the new value of diameter is (1+0.015)DThe new value of shear stress ττ=kTπ D3=kπT D−3=kπ(1−0.03)T∗[(1+0.015)D]−3=kπ(1−0.03)T∗(1+0.015)−3∗D−3=kTπ D3(1−0.03)Using Binomial Theorem for (1+0.015)−3(1+0.015)−3=1−3+(−3)∗0.015+−3∗(−3−1)2!∗0.0152+−3∗(−3−1)∗(−3−2)3!∗0.0153=1−0.045+1.3τ=kTπ D3(1−0.03)∗0.956=0.927kTπ D3This percentage of error (1−0.927)∗100=7.3%which means that the new value of shear stress τis 7.3% too small.Now try to solve the following example:The energy W stored in a flywheel is given by W=k r5N2, where kis a constant, ris a radius and N, number of revolutions. Determine the approximate change in Wwhen ris increased by 1.3% and Nis decreasedby 2% applying the Binomial Theorem Formula. [2.5% increase]2