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3 years ago

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A damping force affects the vibration of a spring so that the displacement of the spring is given by y = e−4t(cos 2t + 3 sin 2t)

. Find the average val

1 answer:

astraxan [27]3 years ago

4 0

Answer:

Average velocity = dy/dt = 4sin2t(2t+3)-4cos2t(6t+1)e^-4t(cos 2t + 3 sin 2t).

Explanation:

Velocity is defined as the rate of change in displacement.

Velocity = change in displacement/time taken

Given the displacement of the string as;

y = e^−4t(cos 2t + 3 sin 2t).

To get the average velocity, we will find the derivative of the displacement with respect to time.

Using function of a function to solve this;

Let u = 4t(cos 2t + 3 sin 2t)... (1)

y = e^-u... (2)

Differentiating both functions with respect to their variables we have;

dy/du = -e^-u

du/dt is gotten using the product rule to have;

du/dt = 4t(-2sin2t+6cos2t)+4(cos2t+3sin2t)

Opening up the bracket we have;

du/dt = -8tsin2t+24tcos2t+4cos2t+12sin2t

Collecting like terms;

-8tsin2t+12sin2t+24tcos2t+4cos2t

du/dt = -4sin2t(2t-3)+4cos2t(6t+1)

dy/dt = dy/du × du/dt

dy/dt = -e^-u × -4sin2t(2t+3)+4cos2t(6t+1)

Substituting u = 4t(cos 2t + 3 sin 2t) into dy/dt, we will have;

dy/dt = -e^-4t(cos 2t + 3 sin 2t) × -4sin2t(2t+3)+4cos2t(6t+1)

dy/dt = 4sin2t(2t+3)-4cos2t(6t+1)e^-4t(cos 2t + 3 sin 2t).

The average velocity of the wave function therefore give us;

dy/dt = 4sin2t(2t+3)-4cos2t(6t+1)e^-4t(cos 2t + 3 sin 2t).

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A train at a constant 79.0 km/h moves east for 27.0 min, then in a direction 50.0° east of due north for 29.0 min, and then west

ivolga24 [154]

Answer:

Magnitude of avg velocity, $|v_{avg}| = 18.9 km/h$

$\theta' = 56.85^{\circ}$

Given:

Constant speed of train, v = 79 km/h

Time taken in East direction, t = 27 min = $\frac{27}{60} h$

Angle, $\theta = 50^{\circ}$

Time taken in $50^{\circ}$ east of due North direction, t' = 29 min = $\frac{29}{60} h$

Time taken in west direction, t'' = 37 min = $\frac{27}{60} h$

Solution:

Now, the displacement, 's' in east direction is given by:

$\vec{s} = vt = 79\times \frac{27}{60} = 35.5\hat{i} km$

Displacement in $50^{\circ}$ east of due North for 29.0 min is given by:

$\vec{s'} = vt'sin50^{\circ}\hat{i} + vt'cos50^{\circ}\hat{j}$

$\vec{s'} = 79(\frac{29}{60})sin50^{\circ}\hat{i} + 79(\frac{29}{60})cos50^{\circ}\hat{j}$

$\vec{s'} = 29.25\hat{i} + 24.54\hat{j} km$

Now, displacement in the west direction for 37 min:

$\vec{s''} = - vt''hat{i} = - 79\frac{37}{60} = - 48.72\hat{i} km$

Now, the overall displacement,

$\vec{s_{net}} = \vec{s} + \vec{s'} + \vec{s''}$

$\vec{s_{net}} = 35.5\hat{i} + 29.25\hat{i} + 24.54\hat{j} - 48.72\hat{i}$

$\vec{s_{net}} = 16.03\hat{i} + 24.54\hat{j} km$

(a) Now, average velocity, $v_{avg}$ is given:

$v_{avg} = \frac{total displacement, \vec{s_{net}}}{total time, t}$

$v_{avg} = \frac{16.03\hat{i} + 24.54\hat{j}}{\frac{27 + 29 + 37}{60}}$

$v_{avg} = 10.34\hat{i} + 15.83\hat{j}) km/h$

Magnitude of avg velocity is given by:

$|v_{avg}| = \sqrt{(10.34)^{2} + (15.83)^{2}} = 18.9 km/h$

(b) angle can be calculated as:

$tan\theta' = \frac{15.83}{10.34}$

$\theta' = tan^{- 1}\frac{15.83}{10.34} = 56.85^{\circ}$

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4 years ago

How many neutrons does element X have if it’s atomic number is 28 and it’s mass number is 81?

Sauron [17]

Answer:

53

Explanation:

M-A=N

Mass-Atomic #= Neutrons

81-28= 53

3 0

4 years ago

Read 2 more answers

Assume: Moving to the right is positive. A(n) 7.7 g object moving to the right at 22 cm/s makes an elastic head-on collision wit

krek1111 [17]

Answer:

Explanation:

We shall apply the formula for velocity in case of elastic collision which is given below

v₁ = (m₁ - m₂)u₁ / (m₁ + m₂) + 2m₂u₂ / (m₁ + m₂)

m₁ and u₁ is mass and velocity of first object , m₂ and u₂ is mass and velocity of second object before collision and v₁ is velocity of first velocity after collision.

Here u₁ = 22 cm /s , u₂ = - 14 cm /s . m₁ = 7.7 gm , m₂ = 18 gm

v₁ = ( 7.7 - 18 ) x 22 / ( 7.7 + 18 ) + 2 x 18 x - 14 / ( 7.7 + 18 )

= - 8.817 - 19.6

= - 28.4 cm / s

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4 years ago

A cardboard box has four forces acting upon it, as shown in the diagram below. What is the magnitude of the unknown force, Funkn

shutvik [7]

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3 years ago

Answer the following. (a) What is the surface temperature of Betelgeuse, a red giant star in the constellation of Orion, which r

bagirrra123 [75]

Answer:

(a) T = 2987.6 k

(b) T = 19986.2 k

Explanation:

The temperature of a star in terms of peak wavelength can be given by Wein's Displacement Law, which is as follows:

$T = \frac{0.2898\ x\ 10^{-2}\ m.k}{\lambda_{max}}$

where,

T = Radiated surface temperature

$\lambda_{max}$ = peak wavelength

(a)

here,

$\lambda_{max}$ = 970 nm = 9.7 x 10⁻⁷ m

Therefore,

$T = \frac{0.2898\ x\ 10^{-2}\ m.k}{9.7\ x\ 10^{-7}\ m}$

<u>T = 2987.6 k</u>

(b)

here,

$\lambda_{max}$ = 145 nm = 1.45 x 10⁻⁷ m

Therefore,

$T = \frac{0.2898\ x\ 10^{-2}\ m.k}{1.45\ x\ 10^{-7}\ m}$

<u>T = 19986.2 k</u>

6 0

3 years ago