a. The velocity v' of the center of mass of this rod after the collision is v' = m2v/(2(MI + m2))
b. The angular velocity of the rod about its center of mass after the collision is ω = -m2 × v/(4×I_cm)
c. Final kinetic energy / initial kinetic energy = 1/2 + (1/16) × (MI/m2)
The principle of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision. Initially, the rod is at rest, so its momentum is zero.
After the collision, the velocity of mass m2 is -v/2, and its mass is m2. Therefore, its momentum after the collision is -m2v/2.
The center of mass of the system must have the same velocity as the momentum is conserved.
The total mass of the system is M = MI + m2. Thus,
0 = (MI + m2) × v' - m2 × v/2
v' = m2v/(2(MI + m2))
The principle of conservation of angular momentum, the total angular momentum before the collision is equal to the total angular momentum after the collision.
Initially, the rod is at rest, so its angular momentum is zero.
After the collision, the velocity of mass m2 is -v/2, and its distance from the center of mass of the rod is L/2.
The angular momentum of mass m2 about the center of mass of the rod is given by m2 × (L/2) × (v/2).
The angular momentum of the rod about its center of mass is I_cm × ω, where I_cm is the moment of inertia of the rod about its center of mass, and ω is the angular velocity of the rod about its center of mass.
Thus,
0 = 0 + m2 × (L/2) × (v/2) + I_cm × ω
ω = -m2 × v/(4×I_cm)
The initial kinetic energy of the system is given by (1/2)MI0² + (1/2)m2v², which simplifies to (1/2)m2v².
The final kinetic energy of the system is given by (1/2)MIv'² + (1/2)m2(-v/2)², which simplifies to (1/2)(MI + m2)(m2v²)/(4(MI + m2)²) + (1/8)m2v².
Thus,
Final kinetic energy / initial kinetic energy
= [(1/2)(MI + m2)(m2v²)/(4(MI + m2)²) + (1/8)m2v²] / ((1/2)m2v²)
= 1/2 + (1/16) × (MI/m2)
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a. The velocity v' of the center of mass of this rod after the collision is (m₂ × v) / (2 × MI)
b. ω' = 0
c. (final kinetic energy) / (initial kinetic energy) = 0
How did we get the values?To solve this problem, use the principles of conservation of linear momentum and angular momentum.
a. Conservation of linear momentum:
Before the collision:
The initial linear momentum of the system is zero since the rod is at rest.
After the collision:
The final linear momentum of the system is the sum of the linear momentum of the rod and mass m₂.
The linear momentum of the rod can be calculated using its mass (MI) and velocity (v') as MI × v'.
The linear momentum of mass m₂ can be calculated using its mass (m₂) and velocity (-[v / 2]) as -m₂ × [v / 2].
Setting up the conservation of linear momentum equation:
0 = MI × v' - m₂ × [v / 2]
Solving for v':
MI × v' = m₂ × [v / 2]
v' = (m₂ × v) / (2 × MI)
b. Conservation of angular momentum:
Before the collision:
The initial angular momentum of the system is zero since the rod is at rest.
After the collision:
The final angular momentum of the system is the sum of the angular momentum of the rod and mass m2.
The angular momentum of the rod can be calculated using its moment of inertia (MIL²/²) and angular velocity (ω') as (MIL²/² × ω'.
The angular momentum of mass m2 can be calculated using its moment of inertia (0 since it's a point mass) and angular velocity (-[v / (2L)]) as 0.
Setting up the conservation of angular momentum equation:
0 = (MIL²/²) × ω' + 0
Solving for ω':
(MIL²/²) × ω' = 0
ω' = 0
c. Ratio of final kinetic energy to initial kinetic energy:
The initial kinetic energy of the system is zero since the rod is at rest.
The final kinetic energy of the system can be calculated by considering the kinetic energy of the rod and mass m₂.
The kinetic energy of the rod can be calculated using its moment of inertia (MIL²/²) and angular velocity (ω') as (MIL²/²) × (ω')².
The kinetic energy of mass m₂ can be calculated using its mass (m2) and velocity (-[v / 2]) as (m₂ × [v / 2])² / (2 × m₂).
The ratio of final kinetic energy to initial kinetic energy is:
(final kinetic energy) / (initial kinetic energy) = [(MIL²/²) × (ω')² + (m₂ × [v / 2])² / (2 × m₂)] / 0
Since ω' = 0, the numerator becomes 0.
Therefore, the ratio is 0.
In summary:
a. v' = (m₂ × v) / (2 × MI)
b. ω' = 0
c. (final kinetic energy) / (initial kinetic energy) = 0
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The side lengths of a square are each 4p. By adding 3 to the length and subtracting 3 from the width, a rectangle is made. What is the area of the rectangle?
Answer: [tex]16p^{2} -9[/tex]
The area of the rectangle is [tex]16p^{2} -9[/tex].
What is a rectangle?A rectangle is one of the types of quadrilaterals in which all four angles are right angles or equal to 90 degrees. It is four-sided polygon in which the opposite sides are parallel and equal to each other.
For the given situation,
The side length of square = 4p
The length of rectangle, l = 4p+3
The width of rectangle, w = 49-3
The formula of area of rectangle is
[tex]A=lw[/tex]
⇒ [tex]A=(4p+3)(4p-3)[/tex] [∵ [tex](a+b)(a-b)=a^{2}-b^{2}[/tex] ]
⇒ [tex]A=(4p)^{2} -3^{2}[/tex]
⇒ [tex]A=16p^{2} -9[/tex]
Hence we can conclude that the area of the rectangle is [tex]16p^{2} -9[/tex].
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Please help me please
Answer:
D
Step-by-step explanation:
Left to right so it is KM (eliminate B and C). It isn't A because the line doesn't continue to the left through point A, so it is D.
#2 Find the value of x. Round to the nearest degree.
Answer:
49°
Step-by-step explanation:
arccos x = 15/23
Graph and then find the perimeter of this shape. H(3, 0), I(3, 7), J(6, 7), K(6, 0)
Answer:
20 unitsStep-by-step explanation:
Graphed the points to find out it is a rectangle with sides 3 and 7.
Vertical segments with distance 7 - 0 = 7Horizontal segments with the distance 6 - 3 = 3The perimeter is:
P = 2(3 + 7) = 20 unitsAnswer:
50 points
Step-by-step explanation:
I did it and got an A
Which expression is equivalent to (5^3)^-2
1. Bella and Caiden each bought a 12-oz can of soda. They are drinking their sodas at the
rates shown. Who is drinking their soda the fastest?
BELLA
CAIDEN
Min
Oz
0
12.
Ounces
20
18
16
14
12
10
8
6
4
2
2
3
4.5
4
2.
1 2 3 4 5 6 7 8 9 10
Minutes
Answer:
The question is incorrect. Please use a table or a diagram.
PLZ help me i will give u many points
Answer:
im pretty sure its the last one
Step-by-step explanation:
Answer:
be
Step-by-step explanation:
Im giving brainiest to the person who answers it first. Thank you!
Answer: Oscer
Step-by-step explanation:
Write a ratio in three different forms.
Shawn biked 5 miles and ran 2 miles. What is the ratio of the distance Shawn ran to the distance he biked?
Answer:
2:5 is the answer
Step-by-step explanation:
Why is because it says What is the ratio of the distance Shawn ran to the distance he biked shawn ran 2 miles the ratio is 2: and comparing how he rode 5 miles is 2:5
Answer:
2:5
Step-by-step explanation:
im in k12 LOL
fine the equation of the line that goes through the point (2,-9) that is parallel to the line with equation y=-3/4x
Answer:
y=6x
Step-by-step explanation:
g=-9/2
c=0
y=-9/2x
3/4x*-9/2
y=6x
can u guys please help! tysmmmmmm
it's 88.1 x 10 to the power negative 2
and
0,0881
Step-by-step explanation:
88.1÷1000=0,0881
FIND THE AREA
WORK PLEASEEE and thank you
Answer:
Some ways on how to solve the questions:
Use formulas as given by your teacher.
Application of the Pythagorean Theorem would be useful for 5 and 6.
For 7-10 use a formula as given by your teacher (just basic operations like plugging in the formula for those questions)
For 11 and 12 draw it out.
How much water do you add to make 75% sugar syrup using 99g sugar?
Answer: You will need 33ml of water added
Step-by-step explanation:
What amount of tax will be owed on an income of 63,280 if city income tax rate is 1.5% of all income
Answer:
multiply 63280 by 1.5% (.015). The answer is $949.20
Step-by-step explanation:
Which of the following is not a true statement? *
a. The boundary line of y > x + 1 is a solid line.
b. The boundary line of y < 2x - 3 is not a solid line.
c. 3x + y < 2 is an example of linear inequality
d. (1, 1) is a solution of x + 3y > 2
Answer:
b. The boundary line of y < 2x - 3 is not a solid line.
Step-by-step explanation:
It's a linear inequality thus its boundary line must be a solid line.
Joe determined the mass of 2 boxes using a scale. Since the cost of each box is the same, Joe would like to choose the box with the larger mass. Box A Box B 32.012 kg 32.02 kg Place the correct symbol between the 2 masses to help Joe decide. 32.012 Response area 32.02
Answer:
32.012 kg < 32.02 kg
Step-by-step explanation:
32.012 kg < 32.02 kg
Help po plss
1.jane read 10books in one year,while huan rrad 8books.find the ratio jane's to juan's finished books .
2.nadine has 860 female followers out of a total of 1260followes in the social photosharing site find the ratio of female to male followers.
3.a public school has 4415 students and 1200 teachers.what is the ratio of the students to each other teacher.
Answer:
1. 10:8 = 5:4 (simplest form)
2. 860:400 = 43:20 (simplest form)
3. 4415:1200 = 883:240 (simplest form)
If you have 100$ and then lose 25$ ,what is the percent change
Answer:
75 %
Step-by-step explanation:
100$= 100%
1$= 1%
75$ (left)= 75%
answer fast helpp.........................
Answer:
20°
Step-by-step explanation:
In circle with center O, PS is diameter.
[tex] m\angle PRS =90\degree [/tex]
(angle inscribed in semicircle)
PQRS is a cyclic quadrilateral.
Opposite angles of a cyclic quadrilateral are supplementary.
[tex] \therefore m\angle QPS + m\angle QRS = 180\degree [/tex]
[tex] \therefore (x\degree +25\degree) +(45\degree +90\degree) = 180\degree [/tex]
[tex] \therefore x\degree +25\degree + 45\degree +90\degree = 180\degree [/tex]
[tex] \therefore x\degree +160\degree = 180\degree [/tex]
[tex] \therefore x\degree = 180\degree-160\degree [/tex]
[tex]\huge\red {\therefore x\degree = 20\degree} [/tex]
2a+3b=5
b=a-5 help please
Answer:
2a + 3b = 5
b = a - 5
Step-by-step explanation:
2a + 3b = 5
b = a - 5
You can write this another way:
2a + 3(a-5) = 5 ( I added the second formula in the first one )
Now you gotta factor out:
2a + 3a - 15 = 5
Here im gonna do plus 15
2a + 3a = 20 ==> 5a = 20
Now if you divide by 5, you get: a = 4
Now you can fill in the second formula again
b = a - 5 (ill fill this in now)
b = 4 - 5 = -1
So, this makes:
a = 4 , b = -1
Answer:
a = -2
b = -7
Step-by-step explanation:
the best method to solving this system of equations would be substitution since you are given an expression that equals one of the variables you're solving for (which is b = a - 5)
to begin, plug in a - 5 for b in the equation 2a + 3b = 5.
2a + 3b = 5 ⇒ 2a + 3(a - 5) = 5
the next step is to distribute 3 to a - 5. start with distributing 3 to a, which is just multiplying 3 · a. after that, you're left with a minus sign and 5, so that sign gets attached to the 5 and makes it a -5. distribute 3 to -5, which is just multiplying 3 · -5.
2a + 3(a - 5) = 5 ⇒ 2a + 3a - 15 = 5
since you have two terms with the variable a on one side of the equation (2a and 3a), combine the two like terms.
2a + 3a - 15 = 5 ⇒ 5a - 15 = 5
add 15 to both sides of the equation. on the left side, -15 and 15 cancel each other out because -15 + 15 = 0, leaving you with only 5a. on the right side of the equation, you have 5 - 15, which equals -10.
now your equation is 5a = -10.
divide both sides of the equation by 5. on the left side of the equation, you have 5a/5, which equals 1a, or a. on the right side, you have -10/5, which equals -2.
therefore a = -2.
now plug in -2 for a in b = a - 5 in order to solve for b!
b = a - 5 ⇒ b = -2 - 5
subtract -2 - 5.
b = -2 - 5 ⇒ b = -7
therefore b = -7.
your solutions are a = -2 and b = -7 :) i hope this helps! have a great day <3
Which of the following values are solutions to the inequality 6≥4x−5?
Answer:
The solution to the inequality is:
[tex]6\ge \:4x-5\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\le \frac{11}{4}\:\\ \:\mathrm{Decimal:}&\:x\le \:2.75\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:\frac{11}{4}]\end{bmatrix}[/tex]
-5 and -6 are the solution values.
Step-by-step explanation:
Given the inequality
[tex]6\ge \:4x-5[/tex]
Let us solve the inequality
[tex]6\ge \:4x-5[/tex]
switch sides
[tex]4x-5\le \:6[/tex]
Add 5 to both sides
[tex]4x-5+5\le \:6+5[/tex]
Simplify
[tex]4x\le \:11[/tex]
Divide both sides by 4
[tex]\frac{4x}{4}\le \frac{11}{4}[/tex]
Simplify
[tex]x\le \frac{11}{4}[/tex]
Thus, the solution to the inequality is:
[tex]6\ge \:4x-5\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\le \frac{11}{4}\:\\ \:\mathrm{Decimal:}&\:x\le \:2.75\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:\frac{11}{4}]\end{bmatrix}[/tex]
In our case, we are given the values -6, 8, and -5
We have already determined that the solution is x ≤ 2.75.
Since the solution is x ≤ 2.75, so -5 and -6 come under the solution set because -5 and -6 are less than 2.75.
Hence, -5 and -6 are the solution values.
The graph of the solution of the inequality is also attached below.
The solutions to the inequality 6 ≥ 4x − 5 is x ≤ 11/4.
What is inequality?A statement of an order relationship-greater than, greater than or equal to, less than, less than or equal to- between two numbers or algebraic equations.
Now the given inequality is,
6 ≥ 4x − 5
Adding 5 both the side we get,
6 + 5 ≥ 4x − 5 + 5
⇒ 11 ≥ 4x
Dividing both the side by 4 we get,
11/4 ≥ 4x/4
⇒ 11/4 ≥ x
or, x ≤ 11/4
this is the required solution of the given inequality.
Thus, the solutions to the inequality 6 ≥ 4x − 5 is x ≤ 11/4.
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I need help with everything but A. assignment is worth 150 points
please help me if you can please
Answer:
hello! :) have a good day!
Exact Form:
x= 87/35
Decimal Form:
x= 2.48571428
Mixed Number Form:
x= 2 17/35
A student needs 23 pound of modeling clay for a project. If each package has 18 pound of clay, how many packages of clay does the student need for his projec
Answer:
First from 18, you count to 23. So, 19, 20, 21, 22, 23. There were only 5 more pounds needed, so it would be, 1 full package of 18 pounds and 1 quarter of a package (I think), because it is not a full package, another full package of 18 would be 36 pounds. I hope this helped! [It might not]
Write the equation of the line that passes through the points (-4,8) and (-2, -1).
Put your answěr in fully reduced point-slope form, unless it is a vertical or horizontal
line.PLEASEEE HELPPP MEE I’ll give you BRAINLEST
Step-by-step explanation:
find the slope
-1 - 8/ -2 - -4 = -9/2
y - -1 = -9/2(x - -2)
y +1 = -9/2(x+2)
y = -9/2x -10
Answer:
y=-9/2x-10
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(-1-8)/(-2-(-4))
m=-9/(-2+4)
m=-9/2
y-y1=m(x-x1)
y-8=-9/2(x-(-4))
y-8=-9/2(x+4)
y=-9/2x-36/2+8
y=-9/2x-18+8
y=-9/2x-10
Question 1 of 5
Given a sphere with a diameter of 6 cm, find its volume to the nearest whole
number.
A. 113 cm3
B. 37 cm3
C. 85 cm3
D. 905 cm3
Answer:
A. 113cm³
Step-by-step explanation:
Finding the volume with diameter,
[tex]v = \frac{1}{6 }\pi {d}^{3} [/tex]
[tex]v = \frac{1}{6} \times \frac{22}{7} \times {6}^{3} [/tex]
[tex]v = 113.14cm^{3} [/tex]
to the nearest whole number
[tex]v = 113 {cm}^{3} [/tex]
Answer:
113 cm^3
Step-by-step explanation:
I Just took the quiz:)
plzzzzzzzzzzz help.
Answer:
D. The area of the triangle is 9 square feet, because the area of the rectangle is 18 square feet.
Step-by-step explanation:
Please help!! Given that triangle ABC is congruent to triangle DEF, the measure of angle A = 70°, the measure of angle B = 60°, the measure of angle C = 50°, the measure of angle D = (3x + 10)°, the measure of angle E = (y/3 + 20)°, and the measure of angle F = (z^2 + 14)°, find the values of x, y, and z.
Answer:
Ok so since the triangles are congruent, that means that the corresponding angles are congruent. This means angle A = angle D, angle B = angle E, and angle C = angle F.
So:
angle A = angle D
70 = 3x + 10
3x + 10 = 70
3x = 60
x = 20
angle B = angle E
60 = y/3 + 20
y/3 + 20 = 60
y/3 = 40
y = 120
angle C = angle F
50 = z^2 + 14
z^2 + 14 = 50
z^2 = 36
z = 6 (or -6)
Answer:
Step-by-step explanation:
A figure is shown
9 in.
3 in.
2 in.
3 in.
Which expression shows how to find the area of the figure?
A. 9x2 + 3x3
O C. 9+3+2+3
B. 9x 3 + 3 x 2
OD. 9 X5
What is the area of the figure? Enter the answer in the box.
Answer:
C. 9×3 + 3×2
The area of the figure is 33 in.
Step-by-step explanation:
Area is:9x5
Hope i could help u
a block that is 2 inches wide,3 inches long, and 2 inches tall. what is the volume of a tower that is built by 3 blocks?
Answer:
36
Step-by-step explanation:
length*Width*height
3*2*2=12
12*3=36