Given:
a(t) = 6t 4
v(0) = 3 m/s
x(0) = 2 m
t = 5 s
v(t) = a dt = 3t² 4t + C
v(0)=3 C=3
v(t)=3t² 4t + 3
x(t)= v dt = t³ 2t² + 3t + C
x(0)=2 C=2
x(5)=125 50 + 15 + 2 = 92 m
Answer:
x(5) = 92 m

Given: _V (x² + y² + z²) dV, V = sphere of radius R (assume R)
Use spherical coordinates: x²+y²+z² = r², dV = r² sin dr d d
^{2n } d ^n sin d ^R r² * r² dr = 2 n * 2 * ^R r dr = 4n * [r /5]_0^R = 4n R /5
Answer: _V (x² + y² + z²) dV = (4/5) n R

Given: L = 6 m, EI = 3×10^9 N·m², m = 150 kg/m, simply supported
w_n = (n ²/L²)*sqrt(EI/m) = (n ²/6²)*sqrt(3×10^9/150) = (n ²/36)*sqrt(2×10^7)
sqrt(2×10^7) = 4472
w _n = 0.274*4472 = 1225 rad/s
Answer: w_n = 1225 rad/s

Given: ri = 0.1 m, ro = 0.2 m, P_i = 30 MPa, P_o = 0
Radial stress: _r = (P_i*ri² - P_o*ro²)/(ro² - ri²) - (ri²*ro²*(P_o - P_i))/(r²*(ro² - ri²))
Hoop stress: _ = (P_i*ri² - P_o*ro²)/(ro² - ri²) + (ri²*ro²*(P_o - P_i))/(r²*(ro² - ri²))
_r(r): _r = (30*0.01 - 0)/(0.04 - 0.01) - (0.01*0.04*(0 - 30))/(r²*0.03) = 10 - (-12/ r²) = 10 + 12/r² (MPa)
_ (r): _ = 10 + 12/r² (MPa)
Answer: _r(r) = 10 - 12/r² MPa, _ (r) = 10 + 12/r² MPa

Given: I = 80 A, V = 240 V, R_a = 0.2
V_b = V - I*R_a = 240 - 80*0.2 = 240 - 16 = 224 V
Answer: E_b = 224 V