REMARKS In evaluating the sine or cosine
function, the angle is in radians, so you should either set your
calculator to evaluate trigonometric functions based on radian
measure or convert from radians to degrees.
QUESTION If the mass is doubled, is the magnitude
of the acceleration of the system at any position doubled, halved,
or unchanged?
doubled, halved, or unchanged?
PRACTICE IT
Use the worked example above to help you solve this problem.(a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is the following.
x = (0.270 m) cos( * t)
A = | _____ m |
f = | _____ Hz |
T = | _____ s |
(b) Find the maximum magnitude of the velocity and acceleration.
vmax = | ______ m/s |
amax = | ______ m/s2 |
(c) What are the position, velocity, and acceleration of the object
after 1.20 s has elapsed?
x = | ______ m |
v = | ______ m/s |
a = | ______ m/s2 |
EXERCISE
If the object-spring system is described by x = (0.345 m) cos (2.00t), find the following.(a) the amplitude, the angular frequency, the frequency, and the period
A = | ______ m |
? = | ______ rad/s |
f = | ______ Hz |
T = | ______ s |
(b) the maximum magnitudes of the velocity and the acceleration
vmax = | ______ m/s |
amax = | ______ m/s2 |
(c) the position, velocity, and acceleration when t = 0.250 s
x = | ______ m |
v = | ______ m/s |
a = | ______ m/s2 |
x = 0.270 cos(pi t / 8)
A = 0.280 m
w = 2 pi f = pi / 8
f = 0.0625 Hz
(B) Vmax = Aw = (0.270)(pi/8)
= 0.106 m/s
a_max = A w^2 = 0.042 m/s^2
(C) x = 0.270 cos(pi t / 8)
x = 0.24 m
v = 0.048 m/s
a = 0.0374 m/s^2
-----------------------Exrecixe
x = 0.345 cos(2t)
A = 0.345m
w = 2 pi f = 2 rad/s
f =0.318 Hz
(B) Vmax = Aw = (0.345)(2)
= 0.69 m/s
a_max = A w^2 = 1.38 m/s^2
(C) x = 0.345 cos( 2 t)
x = 0.303 m
v = 0.33 m/s
a = 1.21 m/s^2
REMARKS In evaluating the sine or cosine function, the angle is in radians, so you should...
PRACTICE IT Use the worked example above to help you solve this problem. (a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is the following. * = (0.235 m) cos( A = T = (b) Find the maximum magnitude of the velocity and acceleration. "max = m/s a max = m/s2 (c) What are the position, velocity, and...
EXERCISE HINTS: GETTING STARTED I'M STUCK! If the object-spring system is described by x = (0.315 m) cos (1.00t), find the following. (a) the amplitude, the angular frequency, the frequency, and the period rad/s Hz (b) the maximum magnitudes of the velocity and the acceleration max = m/s m/s2 a max = (c) the position, velocity, and acceleration when : = 0.250 s x = v = m/s m/s2
EXAMPLE 13.6 The Vibrating Object-Spring System GOAL Identify the physical parameters of a harmonic oscillator from its mathematical description PROBLEM (a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is * - (0.250 m) cos( 1) (b) Find the maximum magnitude of the velocity and acceleration. (c) What are the position, velocity, and acceleration of the object after...
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Find the amplitude, frequency and period of motion for an object vibrating at the of a horizontal spring if the equation for its position as a function of time is: X = (0.250m) cos(pi/8.00 t) Amplitude Frequency Period Find the position, velocity and acceleration at t = 1.0 s for the wave in question 2: position
A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.30 N is applied. A 0.520-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive. Use the exact values you enter to make later calculations.) (d)...
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