Problem 7: Consider the forced spring-mass system:
d^2y/dt^2 + 5y = 8sin(at) where a is a parameter.
a For which value of a does the system exhibit resonance?
b Find the general solution for the a found in (a).
Answers
a) For a ≈ 0.3979, the system exhibits resonance.
b) The general solution is:
y(t) = c1cos(sqrt(5)t) + c2sin(sqrt(5)t) - 8/(5*cos(sqrt(5)t))
a) The system will exhibit resonance when the frequency of the external force matches the natural frequency of the system. The natural frequency of the system is given by:
ω = sqrt(k/m)
where k is the spring constant and m is the mass of the system. In this case, we have a forced spring-mass system with no damping, so k/m = 5, and ω = sqrt(5). The frequency of the external force is given by f = a/2π. Resonance occurs when ω = 2πf, or:
sqrt(5) = 2πa
Solving for a, we get:
a = sqrt(5)/(2π) ≈ 0.3979
So the system will exhibit resonance when a ≈ 0.3979.
b) To find the general solution, we first find the homogeneous solution by setting the right-hand side of the differential equation to zero:
d^2y/dt^2 + 5y = 0
The characteristic equation is r^2 + 5 = 0, which has roots r = ±sqrt(-5). Since the roots are complex, we can write the homogeneous solution as:
y_h(t) = c1cos(sqrt(5)t) + c2sin(sqrt(5)t)
where c1 and c2 are constants determined by the initial conditions.
To find a particular solution for the non-homogeneous equation, we guess a solution of the form:
y_p(t) = Asin(at) + Bcos(at)
Taking the derivatives of y_p and substituting into the differential equation, we get:
-a^2Asin(at) - a^2Bcos(at) + 5Asin(at) + 5Bcos(at) = 8sin(at)
Equating coefficients of sin(at) and cos(at), we get:
(5 - a^2)A = 0
-a^2B = 8
Since we are looking for a non-trivial solution (A and B are not both zero), the second equation gives us:
B = -8/a^2
Substituting this into the first equation and solving for a, we get:
a = sqrt(5)
So the particular solution is:
y_p(t) = -8/(5*cos(sqrt(5)t))
The general solution is then the sum of the homogeneous and particular solutions:
y(t) = y_h(t) + y_p(t) = c1cos(sqrt(5)t) + c2sin(sqrt(5)t) - 8/(5*cos(sqrt(5)t))
where c1 and c2 are constants determined by the initial conditions.
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Related Questions
Solve by using a system of two equations in two variables.
Six years ago, Joe Foster was two years more than five times as old as his daughter. Six years from now, he will be 11 years more than twice as old as she will be. How old is Joe ?
Answers
Answer:
Joe is 43 years old.
Step-by-step explanation:
Let x be the age of Joe Foster at present
Let y be the age of his daughter at present
Six years ago, their ages are:
x - 6 and y - 6 respectively
Six years from now, their ages will be:
x + 6 and y + 6
Six years ago, Joe Foster was two years more than five times as old as his daughter.
(x - 6) = 5(y-6) + 2
Simplify
x - 6 = 5y - 30 + 2
x = 5y -30 + 2 + 6
x = 5y - 22 ---equation 1
Six years from now, he will be 11 years more than twice as old as she will be.
(x + 6) = 2(y+6) + 11
Simplify
x + 6 = 2y + 12 + 11
x = 2y + 12 + 11 -6
x = 2y + 17 ----equation 2
Subtract equation 2 from equation 1
x = 5y - 22
-(x = 2y + 17)
0 = 3y - 39
Transpose
3y = 39
y = 39/3
y = 13
Substitute y = 3 to equation 1 x = 5y - 22
x = 5(13) - 22
x = 65 - 22
x = 43
6, 10, 14, ...
Find the 30th term.
Answers
Answer:
122
Step-by-step explanation:
Four multiplied by 29 add six
The probability of rolling a number cube and getting a 6 and 3.
Answers
We have to find out the probability of getting 6 and 3,
Since this only one roll it is not possible to get 2 numbers in one roll
So, I will presume you want the probability of getting 6 or 3,
Total number of outcomes= {1,2,3,4,5,6} = 6 outcomes
Number of favorable outcomes = {3,6} = 2 outcomes
Probability of rolling 6 or 3 = \(\frac{Number'of 'favorable' outcomes}{Total' number 'of' outcomes}\)
= 2/6
= 1/3 or 0.33333%
Therefore the probability of rolling 6 or 3 is one in three chances
Happy to help :)
If u have any doubts or questions , feel free to ask
in a family with 3 children, excluding multiple births, what is the probability of having exactly 1 girl? assume that having a boy is as likely as having a girl at each birth.
Answers
Using the binomial distribution, it is found that there is a 0.375 = 37.5% probability of having exactly 1 girl.
For each children, there are only two possible outcomes. Either it is a boy, or it is a girl. The probability of a children being a girl is independent of any other children, which means that the binomial distribution is used to solve this question.
Binomial probability distribution
\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)
\(C_{n,x} = \frac{n!}{x!(n-x)!}\)
The parameters are:
x is the number of successes. n is the number of trials. p is the probability of a success on a single trial.In this problem:
3 children, thus \(n = 3\)Equally as likely to be a girl or a boy, thus \(p = 0.5\).The probability of exactly 1 girl is P(X = 1), thus:
\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)
\(P(X = 1) = C_{3,1}.(0.5)^{1}.(0.5)^{2} = 0.375\)
0.375 = 37.5% probability of having exactly 1 girl.
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NEED HELP with these two pages please help me (30 points)
Answers
Answer:
1. B
2. E
3. D
4. B
5. C
6. B
7. E
8. D
9. D
10.D
11. E
please help me out .
ill mark brain list
Answers
Answer:
The amount of time Jill spent to do her homework
= 9 1/2 hours. which equals to 570 minutes.
Which means, the amount of times Jack needed to do his is
= 570 minutes x 1.5
= 855 minutes which equals to 14 1/4 hours
hope this helps
Answer:
Step-by-step explanation:
1) Time spent by Jack = Time spent by Jill * 1 1/2
\(= 9\frac{1}{2}*1\frac{1}{2}\\\\=\frac{19}{2}*\frac{3}{2}\\\\=\frac{19*3}{2*2}\\\\=\frac{57}{4}\\\\=14\frac{1}{4} hours\)
2) Time spent by Jill = Time spent by Jack ÷ 1 1 /2
= 18 1/4 ÷ 1 1/2
= 73/ 4 ÷ 3/2
\(= \frac{73}{4} * \frac{2}{3}\\\\=\frac{73}{2}*\frac{1}{3}\\\\=\frac{73*1}{2*3}\\\\=\frac{73}{6}\\\\=12\frac{1}{6} hours\)
3) Third number = Sum of three numbers - Sum of two numbers
Sum of two numbers = \(8\frac{2}{3}+8\frac{1}{6}\\\)
\(= \frac{26}{3} +\frac{49}{6}\\\\=\frac{26*2}{3*2}+\frac{49}{6}\\\\=\frac{52}{6}+\frac{49}{6}\\\\=\frac{52+49}{6}\\\\=\frac{101}{6}\\\)
Third number = 22 8/15 - 101/6
\(= \frac{338}{15}- \frac{101}{6}\\\\= \frac{338*2}{15*2}- \frac{101*5}{6*5}\\\\= \frac{676}{30}- \frac{505}{30}\\\\= \frac{676-505}{30}\\\\= \frac{171}{30}\\\\=5 \frac{21}{30}\\\\= 5 \frac{7}{10}\)
Como expresar este ejercisio por cada 6 cuadrados hay 3 círculos.
Answers
The ways that the exercise can be expressed such that for every 6 squares there are 3 circles include:
Ratio FormProportional statement Equation form How to express the exercise ?This question is asked in Spanish on an English site so the answer will be provided in English for better learning by other students.
The ratio of squares to circles is 6:3 or simplified, 2:1. This means for every 2 squares, there is 1 circle.
You could also use a proportional statement such that the number of squares is twice the number of circles. For every 6 squares, there are 3 circles.
There is also equation form where we can say, if S is the number of squares and C is the number of circles, the relationship could be expressed as S = 2C. This means the number of squares is twice the number of circles.
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The translated question is:
How to express this exercise for every 6 squares there are 3 circles.
the manufacturer of claritin, a drug for allergy sufferers that was designed to not make them sleepy, gave physicians free samples of the drug because:
Answers
Claritin, a medication for allergy patients that was made to prevent drowsiness, was given away as free samples to doctors by its maker because of following reasons
1) One method of performing advertising is by giving samples to doctors. The business is not required to provide further product information in this type of advertisement. Samples of the product are available for customers or audiences to immediately use and evaluate the product's quality and suitability. Consequently, choosing this alternative is the best choice.
2) A common antihistamine is loratadine (Claritin). It inhibits histamine, a substance your body naturally produces that may cause allergy symptoms including itchy eyes.
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Consider the regression model below and let (X i ,Y i ),i=1,…,n be an i.i.d. set of observations. Y i =β 0 +β 1 X i +u i. Specify the order of the 3 steps required to test the null hypothesis that the slope coefficient β 1 equals zero? That is, H 0 :β 1 =0 A. Compute the standard error of the estimated slope coefficient β 1 act ,SE( β 1act )= σ^ β^ 1 . B. Compute the p− value, Pr H 0 (∣t∣>∣t act ∣). (Enter 1, 2 or 3 in each box)
Answers
The order of the three steps required to test the null hypothesis that the slope coefficient β₁ equals zero in the regression model is as follows:
1. Compute the standard error of the estimated slope coefficient β₁.
2. Calculate the test statistic (t-statistic).
3. Compute the p-value.
To test the null hypothesis that the slope coefficient β₁ equals zero in the regression model, we need to follow the three steps in the specified order:
1. Compute the estimated standard error of the slope coefficient β₁ (SE(β₁)):
The standard error measures the precision of the estimated slope coefficient. It indicates the average distance between the estimated β₁ and the true β₁. To compute SE(β₁), we need to estimate the error term's standard deviation (σ) and use the formula:
SE(β₁) = sigma / √(Σ(Xᵢ - X_cap)²),
where sigma is the estimated standard deviation of the error term and Xᵢ represents the observed values of the independent variable.
2. Calculate the test statistic (t-statistic):
The t-statistic measures the deviation of the estimated slope coefficient from the hypothesized value (zero in this case) in terms of standard errors. It is computed as:
t-act = (β₁ - 0) / SE(β₁).
Here, t-act represents the actual value of the t-statistic.
3. Compute the p-value:
The p-value is the probability of obtaining a t-statistic as extreme as or more extreme than the observed value (|t-act|) under the null hypothesis. To calculate the p-value, we compare the observed t-act with the t-distribution's critical values using the degrees of freedom (n - 2, where n is the number of observations). The p-value is given by:
p-value = Pr(H₀)(|t| > |t-act|).
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150-5r>87.5 Il mark brainliest
Answers
Answer:
r < 12.5
Step-by-step explanation:
Given
150 - 5r > 87.5 ( subtract 150 from both sides )
- 5r > - 62.5
Divide both sides by - 5, reversing the symbol as a result of dividing by a negative quantity.
r < 12.5
Answer:
r<12.5
Step-by-step explanation:
First, we need to isolate the variable so we subtract 150 from both sides.
-5r>-62.5
we then divide by -5 to get the value of x alone. Since this is an inequality and we are dividing by a negative we flip the inequality symbol. Remember to always flip the inequality symbol when dividing by a negative number.
R<12.5
Menaha traveled 86km 520m by train and 11km 480m by car What ditance did he travel in all?
Answers
In total, Menaha traveled 97km 1000m (97.1km).
What is distance?Distance is a numerical measurement of how far apart two objects, points, or places are in space. Distance can be measured in linear units such as meters, kilometers, feet, miles, etc. It can also be measured in angular units such as degrees or radians.
Distance can also refer to the space between two points in time, such as the time between two events. Distance can be used to measure physical distance, time, or even emotional distance.
To calculate this, the two distances must be added together.
The train distance is =86km 520m (86.52km)
and the car distance is =11km 480m (11.48km).
When added together, =86km 520m+11km 480m = 97.52km.
However, since the distances are measured in km and m,
it is necessary to convert the measurements into a single unit of measurement.
To do this, the measurements must be converted into metres.
The train distance is 86,520 metres
And the car distance is 11,480 metres.
When added together,
the total distance is 97,000 metres (97km).
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Fill in the blank. I NEED AN ANSWER FASTTT
Answers
Answer:
It is =
Step-by-step explanation:
Because (6 x 1 = 6 ) + 1 x 1/100 = 6.1
what is 1 oz in tbsp
Answers
There are 2 tablespoons in 1 fluid ounce. This conversion is often used in cooking and baking recipes, where ingredients are measured in ounces or tablespoons.
Fluid ounces and tablespoons are both units of volume used to measure liquids in cooking and baking recipes. One fluid ounce is equal to 29.5735 milliliters or approximately 2 tablespoons, which is equivalent to 6 teaspoons. Therefore, there are 2 tablespoons in 1 fluid ounce.
This conversion is important to know when following recipes that call for ingredients in fluid ounces or tablespoons. It allows for accurate measurement of ingredients, which is crucial for successful cooking and baking. Other common units of volume used in the kitchen include teaspoons, cups, and quarts, among others.
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11 + 16 + 4 = ????????????
Answers
Answer:
31! :D
Step-by-step explanation:
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If you want to join you can,
Good luck!!
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18^9 ÷ 18 ^4 equivalent expression
Answers
Answer:
18^5
Step-by-step explanation:
18^9 ÷ 18^4
= 18 ^ (9-4)
= 18 ^ 5
A student earned a grade of 80%mon a math test that had 20 problems. How many problems on this test did the student answer correctly (Round to the nearest whole number)
Answers
Answer:
16
Step-by-step explanation:
20*80/100 use this formula to find it
The formula for the volume, V, of a cone having the radius, r, and the height, h, is shown below. V=1/3πR^H Write the formula to calculate the height, H. PLEASE HELP
Answers
Answer:
\(h=\frac{1}{3} \pi r^{2} v\)
Step-by-step explanation:
height is equal to one third times pi times radius squared times volume
The expression of the volume of the cone in terms of height H will be as H = 3V/(πR²).
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
An expression is a mathematical proof of the equality of two mathematical expressions.
As per the given volume of the cone,
V=(1/3)πR²H
Manipulate the above formula as,
[V (3/1)]/(πR²) = H
H = 3V/(πR²)
Hence "The expression of the volume of the cone in terms of height H will be as H = 3V/(πR²)".
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7. Use the Composite Trapezoidal rule with the indicated values of \( n \) to approximate the following integrals. (1 mark) (a) \( \int_{1}^{2} x \ln x d x, \quad n=4 \) (b) \( \int_{2}^{2} x^{3} e^{x
Answers
The Composite Trapezoidal rule is used to approximate the given integrals. In part (a), the integral \(\( \int_{1}^{2} x \ln x \, dx \)\) is approximated using \(\( n = 4 \)\)subintervals. In part (b), the integral\(\( \int_{2}^{2} x^{3} e^{x} \, dx \)\) is given with incorrect limits, so it cannot be evaluated.
To approximate \(\( \int_{1}^{2} x \ln x \, dx \)\) using the Composite Trapezoidal rule, we divide the interval \(\([1, 2]\) into \( n = 4 \)\) subintervals. The step size, \(\( h \)\), is calculated as\(\( h = \frac{b-a}{n} = \frac{2-1}{4} = \frac{1}{4} \)\). Then, we evaluate the function \(\( x \ln x \)\)at the endpoints of each subinterval and sum the areas of the trapezoids formed. The approximation formula for the Composite Trapezoidal rule is: \(\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right]\]\)
Using this formula, we can calculate the approximation for the given integral. The limits of the integral \(\( \int_{2}^{2} x^{3} e^{x} \, dx \)\) are given as \(\( 2 \)\) to 2 which indicates an interval of zero length. In this case, the integral cannot be evaluated since there is no interval over which to integrate.
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Can someone help me pleaseee!
Answers
True or false? If you took a true "il then statement and revered the clauses,
the new statement would also be true
O A True
O B. False
Answers
the answer to ur question is false
Xyz corporatio nis selling 100000 shares of common stock through an underwriter at $15 per share, xyz corporation will receive?
Answers
The XYZ corporation will receive $1500000 for all the shares sold.
What is unitary method?A single unit's value can be determined from the values of multiple units, and multiple units' values can be determined from the values of single units using the unitary method.
Now,
Given: Cost of 1 share = $15
To find: Money received on selling 100000 shares.
Finding:
By unitary method,
Money received on selling 1 share = $15
Money received on selling 100000 shares = 15(100000) = $1500000
Hence, The XYZ corporation will receive $1500000 for all the shares sold.
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What is the surface area
of this square-based
pyramid in square yards?
Answers
Answer:
105ft^2
Step-by-step explanation:
5×5 will equal the area of the base =25ft^2
4 triangles × (area of triangle)
area of triangle is height times base/2
4×5×8/2=80 ft^2
+25=105ft^2
American units omg lol just use Si when will your convert
Sam takes out a $167,000 mortgage for 20 years. He makes monthly payments and at the end calculates that he has paid $240,141 towards the mortgage. What is the approximate APR?
3%
4%
6%
7%
Answers
Answer:
About 3%
Step-by-step explanation:
3% of $167,000 is $5,010
5010 times the 20 years is 100200 add that to the 167000 and you get $267,200
The actual answer is about 2.4% but 3% is the closest.
Which answer choice has the set of numbers ordered correctly from least to greatest
Its 1.71, 1 3/4, pie, 16/9
Answers
Answer: 1.71, 1 3/4, 16/9, pie
Step-by-step explanation: We convert all the numbers to improper fractions. 1.71 = 171/100. 1 3/4 = 175/100 because 3/4 is 0.75. pi is 3.14... which is larger than all of the other numbers. So, pi is the largest. For 16/9, we can just divide to find the solution. Once you divide, you will notice that the answer is 1.77777777... This means that it is larger than 1.71 and 1.75 so 16/9 is the second largest. 1.71 is smaller than 1.75 so the order is 1.71, 1 3/4, pie, 16/9
_________________ is the angle of tilt below the imaginary horizontal plane oriented 90-degrees to the _______________ direction.
Answers
The missing terms in the sentence are "Inclination" and "Vertical."
Inclination is the angle of tilt below the imaginary horizontal plane oriented 90-degrees to the vertical direction. It refers to the angle at which an object or surface deviates from being perfectly horizontal or flat. It measures the slope or tilt of the object or surface in relation to the vertical direction.
The vertical direction is the imaginary line or axis that points directly up or down, perpendicular to the horizontal plane. It is oriented at a 90-degree angle to the horizontal direction, which is typically considered the direction parallel to the ground or any other reference plane.
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Which of these expressions is equivalent to 30b2?
A 3b + 10b
B 3b. 10b
c9b +21b
D 9b21b
Answers
Answer:
B) 3b. 10b
Step-by-step explanation:
B) 3b. 10b = (3x10)(bxb) = 30b²
Pt.2 Please show your work
Find the slope when given the following points
(-3, 2) and (1, 0)
Answers
Answer:
slope is = -1/2
Step-by-step explanation:
y2-y1= 2-0= 2
x2-x1= -3-1= -4
2/-4= -1/2
m=___
x-x
it would be 2-0 OVER (fraction formula shown above) -3-1
so your end fraction would be 2/-4 which would be -0.5
so, your slope would be -0.5
What is the slope of the line?
(Look at the image)
Answers
The slope of the line is 3
When doing slope, it is rise over run. Look at 2 points that connect and figure how much they rise and how much they run. Rise 3 run 1 = 3/1
Please help me . It's urgent
Answers
Answer:
36
Step-by-step explanation:
i think lmk if i am right
There are 100 cm in 1 m. How many centimetres are there in b m
Answers
Answer:
100b
Step-by-step explanation:
Since 100 cm = 1 meter, we need to multiply b*100 to get the number of centimeters.
Pls, mark brainliest!