Oscillations and harmonic motion

A particle moves along the $x$-axis with equation of motion $x(t)=0.080\cos (10\pi t)$, with $x$ in meters and $t$ in seconds. Determine the period, frequency, maximum velocity and maximum acceleration.

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A particle in harmonic motion with angular frequency $\omega =4.0rad/s$ is at the instant $t=0$ at the position $x_0=0.060\mathrm{m}$ with velocity $v_0=-0.32m/s$. Write the equation of motion in the form $A\cos (\omega t+\phi )$.

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A body of mass $0.80\mathrm{kg}$, attached to an ideal spring and resting on a smooth horizontal plane, oscillates with period $0.40\mathrm{s}$. Determine the spring constant.

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A body hanging from an ideal vertical spring stretches it, at equilibrium, by $2.5\mathrm{cm}$. Determine the period of the small vertical oscillations of the body about the equilibrium position.

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A simple pendulum $1.00\mathrm{m}$ long oscillates with small amplitude on the surface of Mars, where the gravitational acceleration is $3.71m/{\mathrm{s}}^2$. Determine the period.

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A simple pendulum $0.800\mathrm{m}$ long performs small oscillations with a measured period of $1.80\mathrm{s}$. Determine the value of the gravitational acceleration at the place of the measurement.

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A body of mass $0.10\mathrm{kg}$ oscillates on a smooth horizontal plane attached to an ideal spring of constant $250N/m$, with amplitude $4.0\mathrm{cm}$. Determine the mechanical energy and the magnitude of the velocity at the position $x=2.0\mathrm{cm}$.

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A block of mass $1.2\mathrm{kg}$ slides on a smooth horizontal plane with speed $1.5m/s$ and hits an ideal spring of constant $300N/m$ fixed to a wall. Determine the maximum compression of the spring and the duration of the contact.

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A uniform rigid rod of mass $m$ and length $L=1.50\mathrm{m}$ is suspended by one end from a frictionless horizontal pivot. Given that its moment of inertia about the perpendicular axis through its center is $\frac{1}{12}m{L}^2$, determine the period of small oscillations.

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A thin uniform ring of mass $m$ and radius $R=0.25\mathrm{m}$ hangs from a horizontal nail that passes through a point on its rim and is perpendicular to the plane of the ring. Given that the moment of inertia of the ring about the perpendicular axis through its center is $m{R}^2$, determine the period of small oscillations in the plane of the ring.

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In a weakly damped oscillator the amplitude is halved in $4.0\mathrm{s}$. Determine the damping constant $\gamma$ and the fraction of the initial energy that remains after $4.0\mathrm{s}$.

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A body of mass $2.0\mathrm{kg}$ is attached to an ideal spring of constant $128N/m$. Determine the coefficient $b$ of the viscous force for which the damping is critical.

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A body of mass $0.25\mathrm{kg}$, attached to an ideal spring of constant $100N/m$ and subject to a viscous force with $b=0.50kg/s$, is driven by a force $F_0\cos (\Omega t)$ with $F_0=1.0\mathrm{N}$. Determine the resonant angular frequency and the maximum amplitude of the steady state.

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