(a) The length of the pendulum is 3.96 m.
(b) The amplitude is 7.89 m.
(c) The frequency is 0.25 Hz.
(d) The maximum velocity is 12.38 m/s.
(e) The total energy is 9.67 J.
(f) The maximum height is 3.94 m.
The length of the pendulum at the given parameters is determined using the following formula.
[tex]T = 2\pi \sqrt{\frac{l}{gcos(\theta)} } \\\\\frac{T}{2\pi } = \sqrt{\frac{l}{gcos(\theta)} }\\\\\frac{T^2}{4\pi ^2 } =\frac{l}{gcos(\theta)}\\\\l = \frac{gcos(\theta)T^2}{4\pi^2} \\\\l = \frac{(9.8)cos(5)\times (4)^2}{4\pi^2} \\\\l = 3.96 \ m[/tex]
The amplitude of the pendulum is calculated as follows;
mgLcosθ = ¹/₂FA
mgLcosθ = ¹/₂(mg)A
Lcosθ = ¹/₂A
A = 2Lcosθ
A = 2 x 3.96 x cos(5)
A = 7.89 m
[tex]\omega = \sqrt{\frac{g}{l} } \\\\\omega = \sqrt{\frac{9.8}{3.96} } \\\\\omega = 1.57 \ rad/s[/tex]
ω = 2πf
f = ω/2π
f = (1.57) / (2π)
f = 0.25 Hz
The maximum velocity is calculated as follows;
v = Aω
v = 7.89 x 1.57
v = 12.38 m/s
E = mgLcosθ
E = 0.25 x 9.8 x 3.96 x cos(5)
E = 9.67 J
h = Lcosθ
h = 3.96 x cos(5)
h = 3.94 m
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