**P1.1. a)** *x*_{1} = 2 cos(40π*t*) + 0.7958 sin (40π*t*), *x*_{2} = −2 cos(40π*t*) + 2.387 sin(40π*t*) and *x*_{3} = 3 cos(40π*t*) + 3.979 sin(40π*t*) = 4.98 cos(40π*t* −0.925); hence, *x*_{max} = 4.98 cm and *ν*_{max} = 6.26 m/s.

**P1.2. a)** , where , cos and sin .

**P1.3.** , where and .

**P1.4. a)** *F* = −*Kx*, where *K* = 9π^{2}*m* = 44.4 N/m. **b)** *U*_{(P)} = 0.888 × 10^{−2} cos^{2}(3π*t*) J, *U*_{(C)} = 0.00888 sin^{2}(3π*t*) J, *U*_{(T)} = 0.00888 J. **c)** *U*_{(P)} = 22.2 *x*^{2}, *U*_{(C)} = 0.00888 − 22.2 *x*^{2}.

**P1.5.** *F* = −*Kx*; hence .

**P1.6. a)** , where *K* = *K*_{1} + *K*_{2}. **b)** The lengths of the springs are: *a* = (*K*_{1}/*K*)*L*_{1} + (*K*_{2}/*K*)(*L* −*L*_{2}), *b* = (*K*_{2}/*K*)*L*_{2} + (*K*_{1}/*K*)(*L* −*L*_{1}) and ...

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