This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Welcome back She🦋is🎀Rodiah — missed you this week. a) Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force acting on the oscillating body is directly proportional to its displacement from the equilibrium position and always directed towards the equilibrium position. This results in a sinusoidal oscillation. b) Four examples of systems that undergo S.H.M: • A mass attached to a spring. • A simple pendulum oscillating with a small amplitude. • A liquid oscillating in a U-tube. • A vibrating tuning fork. c) Given: Mass m = 10 kg Amplitude A = 5 cm = 0.05 m Frequency f = 50 Hz Displacement x = 3 cm = 0.03 m Step 1: Calculate the angular frequency . The formula for angular frequency is: = 2 f Substitute the given frequency: = 2 (50 Hz) = 100 rad/s ≈ 314.16 rad/s c) (i) Find the maximum velocity. The formula for maximum velocity in SHM is: v_max = A Substitute the values for amplitude and angular frequency: v_max = (0.05 m)(100 rad/s) v_max = 5 m/s v_max ≈ 15.71 m/s The maximum velocity is 15.71 m/s. c) (ii) Find the Kinetic Energy (K.E) at a displacement of 3 cm. The formula for Kinetic Energy at a displacement x is: K.E. = (1)/(2)m^2 (A^2 - x^2) Substitute the given values: K.E. = (1)/(2)(10 kg)(100 rad/s)^2 ((0.05 m)^2 - (0.03 m)^2) K.E. = 5 (10000^2) (0.0025 - 0.0009) J K.E. = 50000^2 (0.0016) J K.E. = 80^2 J K.E. ≈ 789.57 J The Kinetic Energy at 3 cm displacement is 789.57 J. c) (iii) Find the Total Energy. The formula for Total Energy in SHM is: T.E. = (1)/(2)m^2 A^2 Substitute the given values: T.E. = (1)/(2)(10 kg)(100 rad/s)^2 (0.05 m)^2 T.E. = 5 (10000^2) (0.0025) J T.E. = 125^2 J T.E. ≈ 1233.70 J The Total Energy is 1233.70 J. c) (iv) Sketch the energy-distance graph. The energy-distance graph for SHM illustrates how Kinetic Energy (K.E.), Potential Energy (P.E.), and Total Energy (T.E.) vary with displacement x. • The Total Energy is constant and is represented by a horizontal line across the graph. • The Potential Energy is zero at the equilibrium position (x=0) and reaches its maximum value at the extreme positions (x=± A). It follows a parabolic curve, P.E. = (1)/(2)m^2 x^2. • The Kinetic Energy is maximum at the equilibrium position (x=0) and becomes zero at the extreme positions (x=± A). It also follows a parabolic curve, K.E. = (1)/(2)m^2 (A^2 - x^2). The sum of K.E. and P.E. at any given displacement always equals the constant Total Energy. Send me the next one 📸