In the engine of a locomotive, a cylindrical piece known as a piston oscillates in SHM in a cylinder head (cylindrical chamber) with an angular frequency of 180 rev/min. Its stroke (twice the amplitude) is 0.76 m.What is its maximum speed?
Its maximum speed is 7.2 m/s.
A piston of an engine of a locomotive executes a Simple Harmonic Motion. Using the formula of the maximum velocity of a body in oscillation, we can get the speed.
The maximum velocity of a body in oscillation, (i)
Conversion of rev/min to rad/sec,
The first step will be to convert the angular frequency into the units of rad/s from the given rev/min.
Using the formula from equation (ii), we get the angular frequency as:
Also, the maximum displacement of the piston is given as:
For SHM, the maximum speed of oscillations using equation (i) and the given values is given by:
Hence, the value of the maximum speed is 7.2 m/s .
A simple harmonic oscillator consists of a block attached to a spring with k=200 N/m . The block slides on a frictionless surface, with an equilibrium point x=0 and amplitude 0.20 m. A graph of the block’s velocity v as a function of time t is shown in Fig. 15-60 . The horizontal scale is set by . What are (a) the period of the SHM, (b) the block’s mass, (c) its displacement at , (d) its acceleration at , and (e) its maximum kinetic energy.
The vibration frequencies of atoms in solids at normal temperatures are of the order of. Imagine the atoms to be connected to one another by springs. Suppose that a single silver atom in a solid vibrates with this frequency and that all the other atoms are at rest. Compute the effective spring constant. One mole of silver (atoms) has a mass of 108 g.
The center of oscillation of a physical pendulum has this interesting property: If an impulse (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no oscillations are felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the “sweet spot” by athletes), the oscillations due to the impact will sting their hands. To prove this property, let the stick in Fig. simulate a baseball bat. Suppose that a horizontal force (due to impact with the ball) acts toward the right at P, the center of oscillation. The batter is assumed to hold the bat at O, the pivot point of the stick. (a) What acceleration does the point O undergo as a result of ? (b) What angular acceleration is produced by about the center of mass of the stick? (c) As a result of the angular acceleration in (b), what linear acceleration does point O undergo? (d) Considering the magnitudes and directions of the accelerations in (a) and (c), convince yourself that P is indeed the “sweet spot.
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