Equation of SHMs are
"x\u2081= 4\u00d710^{-2} cos2\u03c0(t+1\/8); x\u2082=3\u00d710^{-2} cos2\u03c0(t+1\/4)"
Equation of resultant oscillation,
"y = x\u2081+x_2= 4\u00d710^{-2} cos2\u03c0(t+1\/8)+3\u00d710^{-2} cos2\u03c0(t+1\/4)"
"= [4\\times 10^{-2}cos(2\\pi t + \\frac{\\pi}{4})]+[3 \\times 10^{-2}cos(2\\pi t + \\frac{\\pi}{2})]"
Let "A = 4\\times10^{-2} ; B = 3\\times10^{-2}" to make equation simple.
"y = [Acos(2\\pi t + \\frac{\\pi}{4})]+[Bcos(2\\pi t + \\frac{\\pi}{2})]"
Expanding both cos functions,
"y = A[cos(2\\pi t )cos( \\frac{\\pi}{4}) - sin(2\\pi t )sin( \\frac{\\pi}{4})]+B[cos(2\\pi t)cos(\\frac{\\pi}{2})-sin(2\\pi t)sin(\\frac{\\pi}{2})]"
since, "cos(\\pi \/ 4) = sin(\\pi \/ 4) = \\frac{1}{\\sqrt{2}}; sin(\\pi \/2) = 1;cos(\\pi \/2) = 0"
"y = \\frac{A}{\\sqrt{2}} [cos(2\\pi t) - sin(2\\pi t)] - Bsin(2\\pi t)"
"y = \\frac{A}{\\sqrt{2}}cos(2\\pi t) - ( \\frac{A}{\\sqrt{2}} + B) sin(2\\pi t)"
Let "\\frac{A}{\\sqrt{2}} = Psin\\phi; (\\frac{A}{\\sqrt{2}} + B) = Pcos\\phi"
Then "y = Psin(\\phi) cos(2\\pi t) -Pcos(\\phi )sin(2\\pi t) = Psin(\\phi - 2\\pi t)"
"y = -Psin(2\\pi t - \\phi)"
where, "\\phi = tan^{-1}\\frac{\\frac{A}{\\sqrt{2}}}{\\frac{A}{\\sqrt{2}} + B} = tan^{-1}\\frac{A}{A + \\sqrt{2}B} = tan^{-1}\\frac{4\\times 10^{-2}}{4\\times 10^{-2} + 3\\times 10^{-2}\\sqrt{2}} = 25.89^\\circ"
and
"P = \\sqrt{(\\frac{A}{\\sqrt{2}})^2 + (\\frac{A}{\\sqrt{2}} + B)^2} = 6.47 \\times 10^{-2}"
Hence equation will be,
"y = -6.47 \\times 10^{-2}sin(2\\pi t - \\phi)"
Comments
I am so grateful
Leave a comment