A rotating objects motion can be described by its angle (θ). Given that an objects potential energy is U = 100(1-cosθ) and its kinetic energy is K = 10(dθ/dt)^2, form a second-order differential equation. Note that Total Energy = P + K, and that total energy does not change w.r.t. time.
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This is what I've come up with, but I don't believe it's the right answer,
E=U+K
E= 100(1-cosθ) + 10(dθ/dt)^2
d(E)/dt = d/dt (100(1-cosθ)) + d/dt (10(dθ/dt)^2)
0 = 100sinθ + 20(dθ/dt)(d^2θ/dt^2)
-100sinθ = 20(dθ/dt)(d^2θ/dt^2)
-5sinθ =(dθ/dt)(d^2θ/dt^2)
--What am I doing wrong? Any help is appreciated.
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This is what I've come up with, but I don't believe it's the right answer,
E=U+K
E= 100(1-cosθ) + 10(dθ/dt)^2
d(E)/dt = d/dt (100(1-cosθ)) + d/dt (10(dθ/dt)^2)
0 = 100sinθ + 20(dθ/dt)(d^2θ/dt^2)
-100sinθ = 20(dθ/dt)(d^2θ/dt^2)
-5sinθ =(dθ/dt)(d^2θ/dt^2)
--What am I doing wrong? Any help is appreciated.
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U = 100 - 100 cos θ
dU/dt = 100 sin θ dθ/dt
K = 10(dθ/dt)²
dK/dt = 20 dθ/dt d²θ/dt²
dE/dt = 0
d(U + K)/dt = dU/dt + dK/dt
100 sin θ dθ/dt + 20 dθ/dt d²θ/dt² = 0
( obvious condition dθ/dt ≠ 0 )
d²θ/dt² + 5 sin θ = 0
dU/dt = 100 sin θ dθ/dt
K = 10(dθ/dt)²
dK/dt = 20 dθ/dt d²θ/dt²
dE/dt = 0
d(U + K)/dt = dU/dt + dK/dt
100 sin θ dθ/dt + 20 dθ/dt d²θ/dt² = 0
( obvious condition dθ/dt ≠ 0 )
d²θ/dt² + 5 sin θ = 0