A 490-g block connected to a light spring for which the force constant is 5.80 N/m is free to oscillate on a x= 0 frictionless, horizontal surface. The block is displaced t= 0 5.20 cm from equilibrium and released from rest as in x; = A V; = 0 m the figure. (A) Find the period of its motion. A block-spring system that begins its motion from rest with the block at x = A at t (B) Determine the maximum speed of the block. = 0. In this case, o = 0; therefore, x = A cos ot. (C) What is the maximum acceleration of the block?

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Chapter1: Units, Trigonometry. And Vectors
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(B) Determine the maximum speed of the block.
Use the equation to find v,
(max:
V
max
@A = (3.44 rad/s)(5.20 × 10-2 m)
m/s
(C) What is the maximum acceleration of the block?
w²A = (3.44 rad/s)?(5.20 × 102 m)
m/s?
Use the equation to find a:
a
max
%D
max
(D) Express the position, velocity, and acceleration as functions of time in SI units.
Find the phase constant from the initial
x(0)
= A cos p = A → p = 0
condition that x = A att = 0:
Use the equation to write an expression for
x(t). Use the following as necessary: t
x = A cos(@t + p) =
Use the equation to write an expression for
v(t). Use the following as necessary: t
v = -@A sin(@t + p) =
Use the equation to write an expression for
a = -@ʻA cos(@t + q) =
a(t). Use the following as necessary: t
MASTER IT
HINTS:
GETTING STARTED I I'M STUCK!
Use the equations below. (Note that the direction is indicated by the sign in front of the equations.)
x = (0.0579 m)cos(3.44t + 0.1687)
v = -(0.199 m/s)sin(3.44t + 0.1687)
:-(0.685 m/s?)cos(3.44t + 0.168)
a =
(a) Determine the first time (when t > 0) that the position is at its maximum value.
(b) Determine the first time (when t > 0) that the velocity is at its maximum value.
Transcribed Image Text:(B) Determine the maximum speed of the block. Use the equation to find v, (max: V max @A = (3.44 rad/s)(5.20 × 10-2 m) m/s (C) What is the maximum acceleration of the block? w²A = (3.44 rad/s)?(5.20 × 102 m) m/s? Use the equation to find a: a max %D max (D) Express the position, velocity, and acceleration as functions of time in SI units. Find the phase constant from the initial x(0) = A cos p = A → p = 0 condition that x = A att = 0: Use the equation to write an expression for x(t). Use the following as necessary: t x = A cos(@t + p) = Use the equation to write an expression for v(t). Use the following as necessary: t v = -@A sin(@t + p) = Use the equation to write an expression for a = -@ʻA cos(@t + q) = a(t). Use the following as necessary: t MASTER IT HINTS: GETTING STARTED I I'M STUCK! Use the equations below. (Note that the direction is indicated by the sign in front of the equations.) x = (0.0579 m)cos(3.44t + 0.1687) v = -(0.199 m/s)sin(3.44t + 0.1687) :-(0.685 m/s?)cos(3.44t + 0.168) a = (a) Determine the first time (when t > 0) that the position is at its maximum value. (b) Determine the first time (when t > 0) that the velocity is at its maximum value.
Example 15.1
A Block-Spring System
A 490-g block connected to a light spring for which the
force constant is 5.80 N/m is free to oscillate on a
frictionless, horizontal surface. The block is displaced
x= 0
t= 0
|x; = A
v; = 0
5.20 cm from equilibrium and released from rest as in
wwww
the figure.
(A) Find the period of its motion.
A block-spring system that begins its
motion from rest with the block at x = A at t
= 0. In this case, p = 0; therefore, x = A cos
(B) Determine the maximum speed of the block.
ot.
(C) What is the maximum acceleration of the block?
(D) Express the position, velocity, and acceleration as functions of time in SI units.
SOLVE IT
(A) Find the period of its motion.
Conceptualize Study the figure and imagine the block moving back and forth in simple harmonic motion
once it is released. Set up an experimental model in the vertical direction by hanging a heavy object such
as a stapler from a strong rubber band.
Categorize The block is modeled as a particle in simple harmonic motion.
Analyze Use the equation to find the
k
5.80 N/m
%D
angular frequency of the block-spring
490 x 10
-3
kg
m
system:
= 3.44 rad/s
Use the equation to find the period of the
2л
T = - =
system:
3.44 rad/s
Transcribed Image Text:Example 15.1 A Block-Spring System A 490-g block connected to a light spring for which the force constant is 5.80 N/m is free to oscillate on a frictionless, horizontal surface. The block is displaced x= 0 t= 0 |x; = A v; = 0 5.20 cm from equilibrium and released from rest as in wwww the figure. (A) Find the period of its motion. A block-spring system that begins its motion from rest with the block at x = A at t = 0. In this case, p = 0; therefore, x = A cos (B) Determine the maximum speed of the block. ot. (C) What is the maximum acceleration of the block? (D) Express the position, velocity, and acceleration as functions of time in SI units. SOLVE IT (A) Find the period of its motion. Conceptualize Study the figure and imagine the block moving back and forth in simple harmonic motion once it is released. Set up an experimental model in the vertical direction by hanging a heavy object such as a stapler from a strong rubber band. Categorize The block is modeled as a particle in simple harmonic motion. Analyze Use the equation to find the k 5.80 N/m %D angular frequency of the block-spring 490 x 10 -3 kg m system: = 3.44 rad/s Use the equation to find the period of the 2л T = - = system: 3.44 rad/s
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