Answer:
x(t) = - 6 cos 2t
Explanation:
Force of spring = - kx
k= spring constant
x= distance traveled by compressing
But force = mass × acceleration
==> Force = m × d²x/dt²
===> md²x/dt² = -kx
==> md²x/dt² + kx=0 ------------------------(1)
Now Again, by Hook's law
Force = -kx
==> 960=-k × 400
==> -k =960 /4 =240 N/m
ignoring -ve sign k= 240 N/m
Put given data in eq (1)
We get
60d²x/dt² + 240x=0
==> d²x/dt² + 4x=0
General solution for this differential eq is;
x(t) = A cos 2t + B sin 2t ------------------------(2)
Now initially
position of mass spring
at time = 0 sec
x (0) = 0 m
initial velocity v= = dx/dt= 6m/s
from (2) we have;
dx/dt= -2Asin 2t +2B cost 2t = v(t) --- (3)
put t =0 and dx/dt = v(0) = -6 we get;
-2A sin 2(0)+2Bcos(0) =-6
==> 2B = -6
B= -3
Putting B = 3 in eq (2) and ignoring first term (because it is not possible to find value of A with given initial conditions) - we get
x(t) = - 6 cos 2t
==>
Probably for kind of the same reason that speed is expressed as a
relationship between two units. You know, like miles per hour .
I guess the only reason is because no single unit has been invented
to describe density.
The rate of doing work or using energy would always be expressed
as a relationship between two units ... we would say that the rate of
work is "(so many) joules per second". But the "watt" was invented,
so we can say "(so many) watts" instead.
So I guess you're right. Density could be simpler to describe
if we only had a unit for it. Then we wouldn't have to say "(so many)
grams per cubic centimeter". We would just say "(so many) (new unit)".
Let's try it out:
"Uhhh, pardon me Professor . . . I've been working late in the lab,
and I believe I've identified a new substance, hitherto unknown to
the scientific community, and totally unexpected. In its pure form,
the substance appears to be pink, it smells like butterscotch, and
its density is approximately 27.4 Brianas. I think it's time we published
these findings ... with your name as lead investigator, of course."
I like it !
Answer:
f₂ = 468.67 Hz
Explanation:
A beat is a sudden increase and decrease of sound. The beats are produced through the interference of two sound waves of slightly different frequencies. Now we have the following data:
The higher frequency tone = f₁ = 470 Hz
No. of beats = n = 4 beats
Time period = t = 3 s
The lower frequency note = Frequency of Friend's Trombone = f₂ = ?
Beat Frequency = fb
So, the formula for beats per second or beat frequency is given as:
fb = n/t
fb = 4 beats/ 3 s
fb = 1.33 Hz
Another formula for beat frequency is:
fb = f₁ - f₂
f₂ = f₁ - fb
f₂ = 470 Hz - 1.33 Hz
<u>f₂ = 468.67 Hz</u>
Answer:
B) 4500 Pa
Explanation:
As pressure is force per unit area,
P = F/A
It stands to reason that the smallest pressure for a given force is when it is shared by the largest area.
The possible areas are
0.30(0.40) = 0.12 m²
0.30(0.50) = 0.15 m²
0.40(0.50) = 0.20 m²
The pressure when the face with the largest area (0.20 m²) is down is
P = 900 / 0.20 = 4500 N/m² or 4500 Pa
the other possible pressures would be
900/0.15 = 6000 Pa
900/0.12 = 7500 Pa
which are both larger than our solution.