An object with mass m = 1 kg is attached to a spring with spring constant k and a dashpot with c = 12 . The mass is set in motion with initial position Xo = 1 meter and v, = -2 meters/second. m/s 1a. (5 points) The spring is stretched 0.5 meters by a force of 13.5 N. Find the spring constant k (in units of ). (ignore the dashpot in when finding k.) N m 16. (5 points) Is this damped mass-spring system underdamped, overdamped, or critically damped? Explain using c2 - 4mk. 1c. (10 points) solve the initial value problem x" + 12x' +27x = 0, x(0) = 1,*'(0) = -2 to find the position function (t) of the object t seconds after it is released.
Answers
The spring constant, k = 27 N/m. The system is critically damped. The position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).
(a) Given, m = 1 kg, c = 12 and the spring is stretched 0.5 meters by a force of 13.5 N. We need to find the spring constant k. Let's consider the force applied on the spring, F = -kx
Here, x = 0.5 m and F = 13.5 N. So, 13.5 = -k (0.5) => k = -27 N/m
Since, we cannot have negative spring constant. Therefore, k = 27 N/m.
(b) The given differential equation is x" + 12x' + 27x = 0.
On comparing with the general equation of a damped mass-spring system, we get m = 1 kg, c = 12, k = 27.
The damping factor is defined as β = c/2m = 6.
Using the damping factor we can determine the type of the system. c2 - 4mk > 0 - overdamped systemc2 - 4mk = 0 - critically damped systemc2 - 4mk < 0 - underdamped system In this case, c2 - 4mk = 12^2 - 4(1)(27) = 0
This indicates that the system is critically damped.
(c) The general equation for a damped mass-spring system is x(t) = e-βt(A cos ωdt + B sin ωdt)
Where, A and B are constants that are determined using initial conditions. x(0) = A = 1 and x'(0) = -βA + ωdB/dt = -2 => B = (1/ω) (-βA + 2)
Now, we need to determine ω.ω2 = k/m - β2/4m2 = 27/1 - (12/2)2/4(1)(27) = 3ω = √3
Substitute the value of A, B and ω in the general equation, we get x(t) = e-6t(cos √3t + √3/3 sin √3t)
Hence, the position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).
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Related Questions
CHALLENGE Randi and her sister made balloon animals and sold them for $0.50 each at
a school carnival. They made $48.00. If Randi made twice as many balloon animals as
her sister, how many balloon animals did each girl make? Show your work.
Answers
List what we know Randi made twice what sister did.
So 2X = Randi balloons made
X= Sister balloons made
We also know they made 0.5 for each animal so sales made / price = number sold
2X + X = 48/0.5
3X = 96
X = 32
So Randi made 64 And her sister made 32.
Cause 2X = Randi and X = sister
If y = 4 find slope, X-intercept and y-intercept.
Answers
Answer:
An equation in the form y = mx + b is in the 'slope y-intercept' form where m is the slope and b is the y-intercept. We can rewrite our equation, y = 4, in slope y-intercept form as follows: y = 0x + 4. Here, it is clear that the slope, or m, is zero. Therefore, the slope of the horizontal line y = 4 is zero
Don’t answer unless positive about answer please :)
Answers
if a is a 7×6 matrix, what is the largest possible rank of a? if a is a 6×7 matrix, what is the largest possible rank of a? explain your answers.
Answers
In both cases, the largest possible rank of matrix is 6 because the rank represents the maximum number of linearly independent rows or columns, which cannot exceed the smaller dimension of the matrix.
The largest possible rank of a 7×6 matrix is 6. This is because the rank of a matrix cannot be greater than the number of rows or columns in the matrix. In this case, the maximum number of linearly independent rows or columns in a 7×6 matrix is 6.
Similarly, the largest possible rank of a 6×7 matrix is also 6. This is because the rank of a matrix is equal to the number of linearly independent rows or columns. Since the matrix has 6 rows and 7 columns, at most 6 of them can be linearly independent.
In summary, the largest possible rank of a matrix is equal to the minimum of the number of rows and columns in the matrix.
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Please give the proof process: 2n3 + 3n +10 = Q( n³).
Answers
2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).
To prove that the expression 2n^3 + 3n + 10 is in the set Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3), we need to show that the expression can be written in the form a(n^3) for some constant "a".
Let's start by factoring out the common factor of n^3 from each term:
2n^3 + 3n + 10 = n^3(2 + 3/n^2 + 10/n^3)
Now, let's rewrite the expression as a single term multiplied by n^3:
2n^3 + 3n + 10 = (2 + 3/n^2 + 10/n^3)n^3
Simplifying the expression inside the parentheses:
= (2n^3 + 3n^2 + 10n^3)/n^3
= (12n^3 + 3n^2)/n^3
= 12 + 3/n
ow, we can see that the expression can be written in the form a(n^3), where a = 12 and n^3 = 3/n.
Therefore, we have shown that 2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).
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Phoebe has 72 roses that she is putting into 5 vases. If she puts the same number of roses in each vase, how many roses are left over?
Answers
Answer:
2 roses left over
Step-by-step explanation:
72/ 5 = 14.4
so we know for sure that we can put 14 roses on each vase but no mor than 14 on each vase Some will be left left over
14x5 = 70
72-70 = 2 (2 roses left over)
Tariq is building his dog a bed. He wants the width of his dog's bed to be .75 times as long as the length. If the length is 1.5 meters, how wide will the bed be?
Answers
Answer:
1.125 meters
Step-by-step explanation:
Given
The length of the dogs bed = 1..5meters
If he wants the width of his dog's bed to be .75 times as long as the length, then;
Width of the bed = 0.75 * length of the bed
Width of the bed = 0.75 * 1.5
Width of the bed = 1.125meters
Hence the bed will be 1.125 meters wide
m/4 - 7 = -3 what is the value of m
Answers
Explanation:
The initial expression is:
\(\frac{m}{4}-7=-3\)To know the value of m, we need to solve the equation for m, so adding 7 to both sides of the equation, we get:
\(\begin{gathered} \frac{m}{4}-7+7=-3+7 \\ \frac{m}{4}=4 \end{gathered}\)Then, we can multiply both sides by 4, so:
\(\begin{gathered} \frac{m}{4}\times4=4\times4 \\ m=16 \end{gathered}\)Is A triangle with side lengths 24,32, 42 a right angle?
Answers
Answer:
No its not a right angle.
Please answer CORRECTLY !!!!!! Will mark BRAINLIEST !!!!!!!!
Answers
Answer:
The answer is 7 meters
Step-by-step explanation:
Its 7 meters because if you divide the area by the width you would get the length (49 divided by 7 = 7)
Graph f(x) = log base 3 x-1
Answers
Hello there. To solve this question, we have to remember how to graph a function.
First, remember that the logarithm in any base is an injective function (one-to-one).
We have to find its key-features before graphing it.
These key-features includes: x-intercept, y-intercept (if it exists), its vertical asymptote.
Okay. To find the x-intercept, set the function equal to zero:
\(f(x)=\log_3(x)-1=0\)Adding 1 on both sides, we get
\(\log_3(x)=1\)Apply the property:
\(\log_a(b)=c\Rightarrow b=a^c\)Hence we have
\(x=3^1=3\)So the x-intercept happens at x = 3.
To determine whether or not it has an y-intercept, we check if the limit as x goes to zero from both sides exists and are equal:
\(\begin{gathered} \lim_{x\to0^+}\log_3(x)-1=-\infty \\ \\ \lim_{x\to0^-}\log_3(x)=-\infty \end{gathered}\)In this case, this means we cannot evaluate this function at x = 0, so it doesn't have y-intercept(s).
Next, to determine its vertical asymptote, we check the value of x for which the argument goes to zero, that is
\(x=0\)Therefore it has a vertical asymptote at x = 0.
The graph is then given by:
This is a sketch of the graph of the function.
The final answer to this question is contained in the last option.
Which of the following correctly describes the relationship between a parameter & a statistic?
a) A statistic is calculated from sample data and it's generally used to estimate a parameter.
b) statistics are a group of subjects selected according to the parameters of the study.
c) A parameter is calculated from sample data and is generally used to estimate a statistic.
d) A perimeter and a statistic are not related.
Answers
(a) correctly describes the relationship between a parameter and a statistic.
The correct description is:
a) A statistic is calculated from sample data and is generally used to estimate a parameter.
In statistics, a parameter refers to a characteristic or measure that describes a population, while a statistic is a characteristic or measure calculated from sample data. Statistics are often used to estimate or infer the corresponding parameters of the population. Therefore, option (a) correctly describes the relationship between a parameter and a statistic.
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What is the solution to this system of equations -3x+5y=-2
3x+7y=26
Answers
Answer: -3x+5y=-2 = x=5/3y+2/3
3x+7y=26 = x=-7/3y+26/3
Step-by-step explanation:
Answer:
A (4,2)
Step-by-step explanation:
edge
Simplify the following expression.
Answers
Answer:
option 1
Step-by-step explanation:
you multiply the numerators and the denominators which gets you 4^1/8
An expression is defined as a set of numbers, variables, and mathematical operations. The given exponential function when simplified will result in 4^(1/8). The correct option is A.
What is an Expression?In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
The given exponential expression can be simplified as shown below.
\((4^{\frac14})^{\frac12}\)
Using the exponential property (mᵇ)ˣ = mᵇˣ,
= \(4^{\frac14 \times \frac12}\)
= \(4^{\frac18}\)
Hence, the given exponential expression when simplified will result in 4^(1/8).
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Why
is i= 0.058?
17. A business owner wants to build a reserve fund. To have \( \$ 50,000 \) in the fund at the end of 10 years, the amount the owner will need to invest annually, assuming an annual return of \( 5 \%
Answers
The annual investment amount required to accumulate a reserve fund of $50,000 in 10 years, assuming an annual return of 5%, is approximately $3,720.
To calculate this, we can use the formula for the future value of an ordinary annuity: \[FV = P \times \left( \frac{{(1 + r)^n - 1}}{r} \right),\] where \(FV\) is the future value (desired reserve fund amount), \(P\) is the annual investment amount, \(r\) is the annual interest rate (in decimal form), and \(n\) is the number of years. Rearranging the formula to solve for \(P\), we have: \[P = \frac{{FV \times r}}{{(1 + r)^n - 1}}.\]Substituting the given values into the formula, we get: \[P = \frac{{50000 \times 0.05}}{{(1 + 0.05)^{10} - 1}} \approx 3720.\]Therefore, the business owner would need to invest approximately $3,720 annually to accumulate a reserve fund of $50,000 at the end of 10 years, assuming a 5% annual return.
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Which equation is NOT equivalent to p- 7 = 22? Select all that apply.
A. p-p-7 = 22-p
B. p-7-p = 22 -
C. p - 7 + 22 = 22 - 7
O
D. p - 7 + 7 = 22 - 22
Ep-7 + 7 = 22 +7
Answers
Answer:
c and d
Step-by-step explanation:
i hope this help
Which of the following is a rational number?
square root 97square root 98, square root 99, square root 100
square root 97
square root 98
square root 99
square root 100
Part A: Find a rational number that is between 5.2 and 5.5. Explain why it is rational. (2 points)
Part B: Find an irrational number that is between 5.2 and 5.5. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth. (3 points)
Answers
Answer:
Square root of 100.
Step-by-step explanation:
Step-by-step explanation:
We can write 10 as a fraction 10/1, therefore,\(\sqrt{}\)100 is a rational number.
Part A : A rational no. between 5.2 and 5.5 is 5.3.
It is rational because it can be expressed in the form
p/q where p and q are integers and q is not equal to 0, which is 53/10
Part B: A rational no. between 5.2 and 5.5 is 5.29150262213
An irrational number between 5.2 and 5.5 is 5.29150262213. It is irrational because there is no pattern that repeats and it cant be written as a fraction of two whole numbers.
Answer:
Step-by-step explanation:
Square root of all prime numbers is irrational.
97 is an prime number. So √97 is an irrational number.
√98 = \(\sqrt{2*7*7}=7\sqrt{2}\) is an irrational number.
\(\sqrt{99}=\sqrt{3*3*11} =3\sqrt{11}\) is an irrational number
√100 = 10 is a rational number.
Part A:
5.3 is a rational number.
Rational can be written in p/q form and when divided the result will either terminating decimal or non terminating repeating decimal
Part B:
5.3020050......
Irrational numbers are non terminating non repeating numbers
which of the following give chart elements a realistic, three-dimensional look?A. axisB. chart areaC. legendD. plot area
Answers
The option that gives chart elements a realistic, three-dimensional look is D. Plot Area.
The option that gives chart elements a realistic, three-dimensional look is the plot area. The plot area is the portion of the chart that displays the data points and is typically the main focus of the chart. Adding shading and depth to the plot area, can give the chart a more lifelike appearance and make the data more visually engaging. Additionally, three-dimensional charts can also be created by using specialized charting software that allows for the creation of 3D visualizations. These types of charts can be useful for highlighting patterns and trends in the data, especially when dealing with large datasets.
In a chart, the plot area is the space where the actual data points, bars, lines, or pie slices are displayed. By applying three-dimensional effects to the plot area, you can make the chart elements appear more realistic and visually appealing. This can enhance the viewer's understanding of the data being represented and make the chart more engaging.
In summary, to give chart elements a realistic, three-dimensional look, you should focus on the plot area (Option D). The axis (Option A), chart area (Option B), and legend (Option C) do not directly contribute to creating a three-dimensional appearance for the chart elements.
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Charlie's teacher claims that he does not study and just guesses on exams. on an exam with 201 true-false questions, Charlie answered 53.7% of the questions correctly. Calculations using these results show that if he were really just guessing, there would be roughly one chance and seven that he would do this well. Is there a statistically significant evidence against the teachers claim that Charlie is just guessing? Why or why not?
Answers
Answer:
Step-by-step explanation:
True-False questions consists of 2 possible answer, which are 50% each, so basically is Charlie was just guessing 201 questions, which means his possibility of getting everything correct is \(\frac{1}{2^2^0^1}\), which is really small, his chance of getting 53,7% correct is \(\frac{1}{2^2^0^1 * 0.537}\) which is roughly arround \(\frac{1}{2^2^0^0}\)
PLEASE HELPPP
Answers
Answer:
Step-by-step explanation:
You didnt attatch anything
which graph represents y as a function of x
Answers
Answer:
A
Step-by-step explanation:
The letters A, B, C, and D are used to form four-letter passwords for entering a computer
le. How many passwords are possible if letters can be repeated?
Answers
Step-by-step explanation:
4 positions with 4 possibilities each :
4×4×4×4 = 4⁴ = 256 different possible passwords.
A process by which the investigator assigns the subject to either the treatment or comparison group to improve the likelihood that the two groups will be comparable with regard to known and unknown confounding factors is known as: A. Nonresponse bias B. Blinding C. Randomization D. Stratification
Answers
The correct option is C. Randomization.What is randomization? Randomization is a process that randomly assigns subjects to either a control or an experimental group to reduce bias in scientific research.
Randomization can be done by using various methods such as random number tables, coin flip, and so on.The main objective of randomization is to avoid subjective biases, confounding variables, and statistical outliers in an experiment, to establish cause and effect relationships in a research study.
For instance, in a clinical trial to test the efficacy of a new drug on hypertension, randomization can be performed by dividing the participants into two groups randomly, one group receives the new drug (treatment group), while the other group receives a placebo (control group).The advantage of randomization is that it ensures that known and unknown confounding variables are distributed evenly between the two groups.
Thus, the chances of drawing incorrect conclusions from the research are minimized. It also increases the generalizability and reproducibility of the results obtained.
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Which angles are adjacent to each other? Question is located in the attached file (image)
Answers
Answer:
∠ 1 and ∠2
Step-by-step explanation:
Adjacent angles share a common ray or side between them, and have the same vertex.
Therefore, the adjacent angles from the given image are: ∠ 1 and ∠2.
The other given options are not adjacent angles because they do not have the same common vertex and ray.
PLSSSS HELPPP ME SOMEBODY WITH THESE MATH QUESTIONS!!!!!!!!! OR AT LEAST TRYYYYYYYYYYY. REWARD BRAINLIESTTTTT
Answers
Answer:
i would help but i am very bad at math
Step-by-step explanation:
i’m at sorry
50 Points
What is the best approximation of the solution to the system to
the nearest integer values?
O(7, -2)
O(-2, 6)
O (6, -2)
O(-2,7)
Answers
Answer:
B) (-2, 6)Step-by-step explanation:
The solution of linear system of equations is the point of intersection of the lines.
Closer view of the graph gives us the point ( - 2, 6 ). See the attached.
Add vertical and horizontal lines from the point of intersection and take the closest values to the integer numbers on the x-axis and the y-axis. The numbers are -2 on the x-axis and 6 on the y-axis, therefore the answer is (- 2, 6).
The matching choice is B.
What is the slope of the line?
Answers
Answer:
1/4
Step-by-step explanation:
The slope of a line is rise over run. We can see here rise is 1 and run is 4. 1 over 4 = 1/4.
The ratio of c:d is 1:3. The ratio of d:e is 2:3. What is the ratio of c:d:e?
Answers
The ratio of c:d:e is 1:3:2.
How The answer was obtainedTo find the ratio, you can multiply the first ratio by the second ratio: 1 * 2 = 2 and 3 * 3 = 9, so the ratio becomes 2:9:6, which can be simplified to 1:3:2.
A ratio is a comparison between two values. To calculate a ratio, divide the first value by the second value. For example, if you have three apples and seven bananas, the ratio of apples to bananas would be 3/7.
You can also simplify a ratio by dividing both values by their greatest common divisor. For example, if the ratio of apples to bananas is 6/14, you could simplify it to 3/7.
It's also possible to express a ratio as a fraction, decimal, or percentage. To convert a ratio to a fraction, simply divide the two values. To convert a ratio to a decimal, divide the numerator by the denominator. To convert a ratio to a percentage, multiply the ratio by 100.
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draw a translation of triangle ABC that is 2 units down and 3 units to the left
Answers
Answer: If there is a drawing then try copy that exact drawing but move it 2 units or squares down and 3 units or squares to the left
Step-by-step explanation:
work out the volume of a square based pyramidheight-24.4cmlength-17.2cm
Answers
Answer
=2406.16533
Step-by-step explanation: