Shortcut Methods
Numerical 1:
Period of Oscillation of a Pendulum
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The period of oscillation (T) of a pendulum is given by the formula ( T = 2\pi \sqrt{\frac{L}{g}} ), where (L) is the length of the pendulum and (g) is the acceleration due to gravity.
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Substituting the given values (L = 1) m and (g = 9.8) m/s², we get ( T = 2\pi \sqrt{\frac{1}{9.8}} \approx 2.01 ) seconds.
Numerical 2:
Frequency of Oscillation of a Mass-Spring System
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The frequency of oscillation (f) of a mass-spring system is given by the formula ( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ), where (k) is the spring constant and (m) is the mass attached to the spring.
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Substituting the given values (k = 100) N/m and (m = 2) kg, we get ( f = \frac{1}{2\pi} \sqrt{\frac{100}{2}} \approx 1.59 ) Hz.
Numerical 3:
Maximum Speed of a Mass-Spring System
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The maximum speed ( v_{max} ) of a mass-spring system is given by the formula (v_{max} = A\sqrt{\frac{k}{m}} ), where (A) is the amplitude of oscillation, (k) is the spring constant, and (m) is the mass attached to the spring.
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Since the mass is stretched (1) m from its equilibrium position, (A = 1) m. Substituting the given values (A = 1) m, (k = 100) N/m, and (m = 5) kg, we get (v_{max} = 1\sqrt{\frac{100}{5}} = 4.47 ) m/s.
