Question

A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 130 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. a. Determine the spring constant k. k = 1000 Newtons / meter b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y'', t. Differential equation: 10y''+1000y=130cos(8t) help (equations) Initial conditions: y(0) = 0 and y'(0) = 0 help (numbers) c. Solve the initial value problem for y(t). y(t) = 13/20tsin(8t) help (formulas) d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ? t < ?. If there is no such maximum, enter NONE. maximum excursion = none meters help (numbers)

          A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 130 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.

a. Determine the spring constant k.
k = 1000 Newtons / meter

b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y'', t.

Differential equation: 10y''+1000y=130cos(8t) help (equations)

Initial conditions: y(0) = 0 and y'(0) = 0 help (numbers)

c. Solve the initial value problem for y(t).
y(t) = 13/20tsin(8t) help (formulas)

d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ? t < ?. If there is no such maximum, enter NONE.
maximum excursion = none meters help (numbers)

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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 130 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.

a. Determine the spring constant k.
k = 1000 Newtons / meter

b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y”, t.

Differential equation: 10y”+1000y=130cos(8t) help (equations)

Initial conditions: y(0) = 0 and y'(0) = 0 help (numbers)

c. Solve the initial value problem for y(t).
y(t) = 13/20tsin(8t) help (formulas)

d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ? t < ?. If there is no such maximum, enter NONE.
maximum excursion = none meters help (numbers)

Added by Brenda G.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Neelesh Sharma

03/18/2022

Step 1

The spring stretches 9.8 cm (0.098 m) under the weight of a 10 kg object. The force due to gravity is \( F = mg = 10 \times 9.8 = 98 \) N. Using Hooke's Law, \( F = kx \), we have: \[ 98 = k \times 0.098 \] Solving for \( k \): \[ k = \frac{98}{0.098} = 1000 Show more…

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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 130 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. a. Determine the spring constant k. k = 1000 Newtons / meter b. Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y, y', y'', t. Differential equation: 10y''+1000y=130cos(8t) Initial conditions: y(0) = 0 and y'(0) = 0 c. Solve the initial value problem for y(t). y(t) = 13/20tsin(8t) d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < ∞. If there is no such maximum, enter NONE. maximum excursion = none meters

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Transcript

00:01 In this question, we have a spring system, spring mass system that is vertically hanging.

00:12 Spring mass system that is vertically hanging.

00:15 And we have to find the force is applied onto this from the rest state.

00:24 We have to find the first is the spring constant.

00:29 So we have a spring mass system that is hanging.

00:33 Now because of this spring, it is a spring.

00:36 It is stressed by 9 .8 centimeter.

00:42 Mass is given as 10pc.

00:48 So, we'll directly apply f equals to m .a.

00:53 Equals to kx.

00:55 So, here, k multiplied by x, x is changing the spring length is equal to mk.

01:07 The mass is given as n.

01:09 N mass is so m g so the spring constant will be equals to mass is given as 10 g is 9 .8 this will be divided by 100 square to convert this centimeter and two meters and divided by 1 upon x x is given as x this is 9 .8 only x is given as x is given as 9 .8 only x is given as 9 .8 only x is given as 9 .8 and this is given as 9 .8 this will be divided by 100 so these are the values now we have cancelled out this to this so the final case value will be 1000 newton meter meter meter newton per meter now the next part we have applied a force the value of the forces for this we need to write the differential equation of motion of motion that is mx double dot plus cf dot plus kf is equals to f is the force applied this is the spring constant that is the damping factor which is the mass.

02:48 They will put all the value the mass is given as 10 so this will be y double dot plus this part is zero because there is no 10 things involved then we have thousand at this one and y is equals to 130 plus 80 bit of solving this we'll get the final equation 10y is equal to 13 plus 80.

03:29 Now we have to solve this.

03:32 For solving this differential equation, we need to first need to auxiliary equation.

03:39 What we have? we will write this r square by converting this only.

03:46 R square plus 100 is equal to 0...

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