Concept explainers
(a)
The position of the particle at the end of
(a)
Answer to Problem 11P
The position of the particle at the end of
Explanation of Solution
Given information:
The initial position of the particle is
The formula for the position of the particle is,
Substitute
Conclusion:
Therefore, the position of the particle at the end of
(b)
The velocity of the particle at the end of
(b)
Answer to Problem 11P
The velocity of the particle at the end of
Explanation of Solution
Given information:
The position of the particle is
The formula for the velocity of the particle is,
Substitute
Conclusion:
Therefore, the velocity of the particle at the end of
(c)
The position of the particle in
(c)
Answer to Problem 11P
The position of the particle in simple harmonic motion for
Explanation of Solution
Section 1:
To determine: The angular frequency of the particle.
Answer: The angular frequency of the particle is
Given information:
The position of the particle is
The formula for the acceleration of the particle is,
Substitute
Section 2:
To determine: The amplitude of the motion.
Answer: The amplitude of the motion is
Given information:
The position of the particle is
The general form of position of the particle is,
At the time
Substitute
The general form of velocity of the particle is,
Substitute
Solve the equation (I) and equation (II) to obtain value of
Section 3:
To determine: The phase constant of the motion.
Answer: The phase constant of the motion is
Given information:
The position of the particle is
Substitute
Section 4:
To determine: The position of the particle in simple harmonic motion for
Answer: The position of the particle in simple harmonic motion for
Given information:
The position of the particle is
The formula for the position of the particle is,
Substitute
Conclusion:
Therefore, the position of the particle in simple harmonic motion for
(d)
The velocity of the particle in simple harmonic motion for
(d)
Answer to Problem 11P
The velocity of the particle in simple harmonic motion for
Explanation of Solution
Given information:
The position of the particle is
The general form of velocity of the particle is,
Substitute
Conclusion:
Therefore, the velocity of the particle in simple harmonic motion for
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Chapter 12 Solutions
Principles of Physics
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- In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x=5.00cos(2t+6) where x is in centimeters and t is in seconds. At t = 0, find (a) the position of the piston, (b) its velocity, and (c) its acceleration. Find (d) the period and (e) the amplitude of the motion.arrow_forwardA block of unknown mass is attached to a spring with a spring constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 30.0 cm/s. Calculate (a) the mass of the block, (b) the period of the motion, and (c) the maximum acceleration of the block.arrow_forwardWe do not need the analogy in Equation 16.30 to write expressions for the translational displacement of a pendulum bob along the circular arc s(t), translational speed v(t), and translational acceleration a(t). Show that they are given by s(t) = smax cos (smpt + ) v(t) = vmax sin (smpt + ) a(t) = amax cos(smpt + ) respectively, where smax = max with being the length of the pendulum, vmax = smax smp, and amax = smax smp2.arrow_forward
- A block of mass m is connected to two springs of force constants k1 and k2 in two ways as shown in Figure P12.56. In both cases, the block moves on a frictionless table after it is displaced from equilibrium and released. Show that in the two cases the block exhibits simple harmonic motion with periods (a) T=2m(k1+k2)k1k2 and (b) T=2mk1+k2 Figure P12.56arrow_forwardThe equations listed in Table 2.2 give position as a function of time, velocity as a function of time, and velocity as a function of position for an object moving in a straight line with constant acceleration. The quantity vxi appears in every equation. (a) Do any of these equations apply to an object moving in a straight line with simple harmonic motion? (b) Using a similar format, make a table of equations describing simple harmonic motion. Include equations giving acceleration as a function of time and acceleration as a function of position. State the equations in such a form that they apply equally to a blockspring system, to a pendulum, and to other vibrating systems. (c) What quantity appears in every equation?arrow_forwardAn object of mass m moves in simple harmonic motion with amplitude 12.0 cm on a light spring. Its maximum acceleration is 108 cm/s2. Regard m as a variable. (a) Find the period T of the object. (b) Find its frequency f. (c) Find the maximum speed vmax of the object. (d) Find the total energy E of the objectspring system. (e) Find the force constant k of the spring. (f) Describe the pattern of dependence of each of the quantities T, f, vmax, E, and k on m.arrow_forward
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