Show that the expression for the period of a physical pendulum reduces to that of a simple pendulum if the physical pendulum consists of a particle with mass $m$ on the end of a massless string of length $L$.

Bar AB is rotating clockwise with the constant angular velocity of 20 rad/s. For the position shown, determine the angular accelerations of bars BD and DE.

Bar $A B$ rotates in the counterclockwise direction at $6 \mathrm{rad} / \mathrm{s}$. Determine the angular velocity of bar $B D$ and the velocity of point $D$.

verify that each given function is a solution of the differential equation. y'−2ty=1;y=et2 ∫t0e−s2ds+et2

verify that each given function is a solution of the differential equation. ty'−y=t2;y=3t+t2

The half-life of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that for any radioactive material that decays according to the equation Q'=−rQ,the half-lifeτand the decay rate r satisfy the equation rτ=ln 2.

The system is released from rest in the position shown. Calculate the tension $T$ in the cord and the acceleration $a$ of the $30-\mathrm{kg}$ block. The small pulley attached to the block has negligible mass and friction. (Suggestion: First establish the kinematic relationship between the accelerations of the two bodies.)

The displacement of an oscillating object as a function of time is shown in Fig.. What are the period;

A projectile is launched from point A and has a horizontal range $L_1$. If the coefficient of restitution at B is e, determine the distance $L_2$.

n-Butyl mercaptan is the common name of a foul-smelling substance obtained from skunk spray. It is a thiol of the type RX, where R is an n-butyl group and X is the functional group that characterizes a thiol. Write a structural formula for this substance.

The pH–activity profile for glucose-6-phosphate isomerase indicates the participation of a group with a pKa = 6.7 as a basic catalyst and a group with a pKa = 9.3 as an acid catalyst. Draw the pH–activity profile and identify the amino acids that participate in the catalysis.

Which of the following pH values indicates a solution with the highest concentration of hydrogen ions? A. pH = 1, B. pH = 2, C. pH = 7, D. pH = 14.

An object is undergoing SHM with period 1.200 s and amplitude 0.600 m. At t = 0 the object is at x = 0 and is moving in the negative x-direction. How far is the object from the equilibrium position when t = 0.480 s?

A 1.80-kg connecting rod from a car engine is pivoted about a horizontal knife edge. The center of gravity of the rod was located by balancing and is 0.200 m from the pivot. When the rod is set into small-amplitude oscillation, it makes 100 complete swings in 120 s. Calculate the moment of inertia of the rod about the rotation axis through the pivot.