How do you write 30.0 in scientific notation? ____× 10^_____
2 answers:
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a) [m]
b) 0.033 s
c) -0.152 m/s
Step-by-step explanation:
a)
The force acting on the mass-spring system is (restoring force)
where
k = 9 is the spring constant
y is the displacement
Also, from Newton's second law of motion, we know that
where
m = 1 g = 0.001 kg is the mass
y'' is the acceleration
Combining the two equations,
This is a second order differential equation; the solution for y(t) is
where
A is the amplitude of motion
is the angular frequency
The spring starts its motion from its equilibrium position, this means that y=0 when t=0; therefore, the phase shift must be
So the displacement is
The velocity of the spring is equal to the derivative of the displacement:
We know that at t = 0, the initial velocity is 6 in/s; since 1 in = 2.54 cm = 0.0254 m,
And since at t = 0,
Then we have:
From which we find the amplitude:
So the solution for the displacement is
[m]
b)
Here we want to find the time t at which the mass returns to equilibrium, so the time t at which
This means that
We know already that the first time at which this occurs is
t = 0
Which is the beginning of the motion.
The next occurence of y = 0 is instead when
which means:
c)
As said in part a), the velocity of the mass-spring system at time t is given by the derivative of the displacement, so
where we have
is the angular frequency
is the amplitude of motion
t is the time
Here we want to find the velocity of the mass when the time is that calculated in part b):
t = 0.033 s
Substituting into the equation, we find:
Answer (<u>assuming it can be in slope-intercept form)</u>:
y = -x - 1
Step-by-step explanation:
When knowing the slope of a line and its y-intercept, you can write an equation to represent it in slope-intercept form, or y = mx + b format. Substitute the m and b for real values.
1) First, find the slope of the equation, or m. Pick any two points from the line and substitute their x and y values into the slope formula, . I chose the points (0, -1) and (-1, 0):
Thus, the slope is -1.
2) Now, find the y-intercept, or b. The y-intercept of a line is the point at which the line crosses the y-axis. By reading the graph, we can see that the line intersects the y-axis at the point (0,-1), therefore that must be the y-intercept.
3) Now, substitute the found values into the y = mx + b formula. Substitute -1 for m and -1 for b: