A mass hangs from a vertical spring and performs simple harmonic motion about its…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

0.25\,\text{kg}

mass hangs from a vertical spring and performs simple harmonic motion about its equilibrium position (positive displacement is defined as downward). The oscillation period is

0.60\,\text{s}

. At

t=0\,\text{s}

the mass is

0.020\,\text{m}

below equilibrium and moving upward. At

t=0.15\,\text{s}

the mass is observed to be

0.020\,\text{m}

above equilibrium and still moving upward with the same speed it had at

t=0\,\text{s}

. (a) Use the two observations above to obtain simultaneous equations for the displacement at the two times and solve for both the phase angle

\varphi

(in radians,

0\le \varphi<2\pi

) and the amplitude

A

of the motion. Show all algebraic steps. (b) Write the complete analytical expression for the displacement

x(t)

, including numerical values for

A

, the angular frequency, and

\varphi

obtained in part (a). (c) Determine the speed of the mass at the instant it is

0.010\,\text{m}

above equilibrium and moving downward. Indicate direction in your answer. (d) Describe qualitatively how the symmetry of the displacement values at

t=0\,\text{s}

and

t=0.15\,\text{s}

reflects the phase relationship between the position and velocity functions in simple harmonic motion.

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