15.06.2022

In a given problem ,x= number of springs manufactured during a given day why would x>0 be a valid constraint

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Physics

As you know, a common example of a harmonic oscillator is a mass attached to a spring. In this problem, we will consider a horizontally moving block attached to a spring. Note that, since the gravitational potential energy is not changing in this case, it can be excluded from the calculations. For such a system, the potential energy is stored in the spring and is given by U = 12k x 2 where k is the force constant of the spring and x is the distance from the equilibrium position. The kinetic energy of the system is, as always, K = 12mv2 where m

Step-by-step answer

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a) $E = \frac{1}{2} \cdot k \cdot x^{2} + \frac{1}{2} \cdot m \cdot v^{2}$ , b) Amplitude of the motion is $A = \sqrt{\frac{2\cdot E}{k} }$ , c) The maximum speed attained by the object during its motion is $v_{max} = \sqrt{\frac{2\cdot E}{m} }$ .

Explanation:

a) The total energy of the object is equal to the sum of potential and kinetic energies. That is:

E = K + U

Where:

- Kinetic energy, dimensionless.

- Potential energy, dimensionless.

After replacing each term, the total energy of the object at any point in its motion is:

$E = \frac{1}{2} \cdot k \cdot x^{2} + \frac{1}{2} \cdot m \cdot v^{2}$

b) The amplitude of the motion occurs when total energy is equal to potential energy, that is, when objects reaches maximum or minimum position with respect to position of equilibrium. That is:

E = U

$E = \frac{1}{2} \cdot k \cdot A^{2}$

Amplitude is finally cleared:

$A = \sqrt{\frac{2\cdot E}{k} }$

Amplitude of the motion is $A = \sqrt{\frac{2\cdot E}{k} }$ .

c) The maximum speed of the motion when total energy is equal to kinetic energy. That is to say:

E = K

$E = \frac{1}{2}\cdot m \cdot v_{max}^{2}$

Maximum speed is now cleared:

$v_{max} = \sqrt{\frac{2\cdot E}{m} }$

The maximum speed attained by the object during its motion is $v_{max} = \sqrt{\frac{2\cdot E}{m} }$ .

Physics

Step-by-step answer

P Answered by PhD

Explanation:

a) The total energy of the object is equal to the sum of potential and kinetic energies. That is:

E = K + U

Where:

- Kinetic energy, dimensionless.

- Potential energy, dimensionless.

After replacing each term, the total energy of the object at any point in its motion is:

$E = \frac{1}{2} \cdot k \cdot x^{2} + \frac{1}{2} \cdot m \cdot v^{2}$

b) The amplitude of the motion occurs when total energy is equal to potential energy, that is, when objects reaches maximum or minimum position with respect to position of equilibrium. That is:

E = U

$E = \frac{1}{2} \cdot k \cdot A^{2}$

Amplitude is finally cleared:

$A = \sqrt{\frac{2\cdot E}{k} }$

Amplitude of the motion is $A = \sqrt{\frac{2\cdot E}{k} }$ .

c) The maximum speed of the motion when total energy is equal to kinetic energy. That is to say:

E = K

$E = \frac{1}{2}\cdot m \cdot v_{max}^{2}$

Maximum speed is now cleared:

$v_{max} = \sqrt{\frac{2\cdot E}{m} }$

The maximum speed attained by the object during its motion is $v_{max} = \sqrt{\frac{2\cdot E}{m} }$ .

Physics

As you know, a common example of a harmonic oscillator is a mass attached to a spring. in this problem, we will consider a horizontally moving block attached to a spring. note that, since the gravitational potential energy is not changing in this case, it can be excluded from the calculations.for such a system, the potential energy is stored in the spring and is given byu=12kx2,where k is the force constant of the spring and x is the distance from the equilibrium position.the kinetic energy of the system is, as always,k=12mv2,where m is the mass

Step-by-step answer

P Answered by PhD

A) $E=U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2$

Explanation:

The total energy of the system at any point in the motion is equal to the sum of the elastic potential energy of the spring, U, and of the kinetic energy of the mass, K:

$E=U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2$

where

k is the spring constant

x is the compression/stretching of the spring with respect to its equilibrium position

m is the mass of the block attached to the spring

v is the speed of the block

B) $A=\sqrt{\frac{2E}{k}}$

Explanation:

The amplitude of the motion corresponds to the maximum displacement of the mass-spring system. The displacement of the system, x(t), at time t, for a simple harmonic oscillator is given by

$x=A sin(\omega t + \phi)$

where

A is the amplitude

$\omega=\sqrt{\frac{k}{m}}$ is the angular frequency of the motion

t is the time

$\phi$ is the phase (we can take $\phi=0$ for simplicity)

The amplitude of the motion occurs when the displacement of the motion is maximum: x=A. In terms of energy, the mass-spring system is at its maximum displacement (x=A) when all the mechanical energy of the system is elastic potential energy, so when the kinetic energy is zero:

$K=\frac{1}{2}mv^2=0$

$E=\frac{1}{2}kA^2\\A=\sqrt{\frac{2E}{k}}$ (1)

C) $v_{max}=\omega A$

The maximum speed of the system occurs when the elastic potential energy is zero: U=0 and the kinetic energy is maximum, so:

$U=0\\E=\frac{1}{2}mv_{max}^2$

Due to the law of conservation of the mechanical energy, this energy must be equal to the energy of the system at its maximum displacement (1), so we can write

$\frac{1}{2}kA^2=\frac{1}{2}mv_{max}^2$

and solving for $v_{max}$ we find an expression for the maximum speed:

$v_{max}=\sqrt{\frac{kA^2}{m}}=\sqrt{\frac{k}{m}}A=\omega A$

Mathematics

Neil places £2000 in a bank account that pays 1.5% simple interest per year. How much interest will he earn in 6 years?

Step-by-step answer

P Answered by PhD

SI=(P*R*T)/100

P=2000

R=1.5

T=6

SI=(2000*1.5*6)/100

=(2000*9)/100

=180

Neil will earn interest of 180

Mathematics

You can buy jars of the same jam in 2 sizes large jar:440g for £1.54 small jar:340g for £1.26 By considering the price per gram, work out which jar is the best value for money. Working must be shown

Step-by-step answer

P Answered by PhD

Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram

Mathematics

An ice cream shop offers 5 different flavors of ice cream and 9 different toppings. How many choices are possible for a single serving of ice cream with one topping?

Step-by-step answer

P Answered by PhD

For 1 flavor there are 9 topping

Therefore, for 5 different flavors there will be 5*9 choices

No of choices= 5*9

=45

Mathematics

Deanna gets 3 problems incorrect on a math quiz. Her score is 85%. How many questions are on the quiz?

Step-by-step answer

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