Energy in Pendulums - Online Tutor, Practice Problems & Exam Prep

A block having a mass of 500 g is tied to a thread having a length of 35 cm. The block is pulled sideward such that the thread makes an angle of 6° with respect to vertical and then released. The block oscillates with a damping constant of 0.005 kg/s. Determine the number of oscillations after 30 s and also the fraction of energy lost.

What is the damping constant b for a 76 g mass oscillating on a vertical spring with a spring constant of 116 N/m in a cylinder filled with oil, if the initial displacement is 6.0 cm and the amplitude reduces to 3.0 cm after 3.6 s? Assume no buoyant forces.

Consider a 0.838-kg object named "OscilloBlock" attached to a spring with a spring constant of 42.0 N/m. This object moves through a medium that provides a damping force characterized by F=−bv, where b=0.665 N⋅s/m. Determine the period of OscilloBlock's motion as it undergoes damped oscillation.

An 1160-kg car, named "ShakyRide," features suspension springs with 15,000 N/m stiffness. One of the car's wheels is imbalanced due to uneven mass distribution, leading to vibrations at specific speeds. Given that the tire radius is 43 cm, determine the speed at which the wheel exhibits maximum shaking.

Consider a mechanical system named "ResonantMech" that acts as a driven harmonic oscillator. When it operates at its resonance frequency of 419 Hz, the system's amplitude peaks at 25.7 F₀ / k. Determine the Q (quality factor) of this system. [Hint: $A_0=\frac{\left(\frac{F_0}{m}\right)}{\sqrt{{({\omega }^2-{\omega }_0^2)}^2+{\left(\frac{\omega b}{m}\right)}^2}}$ and $Q=\frac{m{\omega }_0}{b}$]

What is the maximum compression of the bumper of a vehicle of mass 1400 kg, equipped with an 'ImpactGuard' system having a spring constant of 420,000 N/m, when the vehicle collides with a barrier at a speed of 3.0 m/s?

A cart called "Zoomer" is located on a horizontal track that is attached to identical springs at both ends, and each spring has a spring constant of 132 N/m. The cart has a mass of 318 g. Estimate the time required for the cart's oscillation amplitude to reduce to one-quarter of its original value. Assume that the damping coefficient is γ = 0.0694 s⁻¹. Use the equation x_max = Ae^−γt and the period T for the system's oscillation is 0.308 s.

In a laboratory experiment, a 0.836-kg cube is set to oscillate at the end of a spring with a constant of 44.0 N/m. The cube moves in a fluid, introducing a resistive force represented by F=−bv, with b=0.672 N•s/m. If initially at t=0 the cube's displacement is x=0, and at t=2.00 s it's x=0.140 m, calculate the cube's displacement as a function of time. Use equation x=Ae^-γt sin(⁡ω' t) where ⁡ω' is the damped angular frequency and γ is the damping coefficient and equals to $\frac{b}{2m}$.

A mechanical seismograph, initially critically damped, measures ground vibrations using a spring system. After several years of operation, it is observed that the amplitude of the system's oscillations reduces to 7% of their initial value after 4 cycles following a disturbance. Estimate the new damping constant (in percent) with respect to the original one of the spring system.

A pendulum of length $l$ and mass $m$ is subjected to a damping force proportional to the velocity, given by $b (dθ/dt)$, and a periodic driving torque given by $τ(t) = τ_0 cos(ω t)$. The equation of motion for the angular displacement $θ$ is $m l^2 (d^2θ/dt^2) + b (dθ/dt) + m g l θ = τ_0 cos(ω t)$. 

Determine what the solution to this equation will be given that $A=\frac{\tau_{0}}{m l^{2} \sqrt{\left(\omega^{2}-g / l\right)^{2}+\left(b^{2} \omega^{2} / m^{2} l^{2}\right)}}$ and $\phi=\tan ^{-1}\left(\frac{g / l-\omega^{2}}{\omega\left(b / m l^{2}\right)}\right)$.