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posledela
2 years ago
15

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s,

how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall?
Physics
1 answer:
Alekssandra [29.7K]2 years ago
4 0

Answer:

d∅/dt = - 0.18101999999  rad/s

Explanation:

Length of the ladder = 10 ft

distance of the bottom of the ladder to the wall = x

where y = height of the wall

using SOHCAHTOA principle

tan ∅ = opposite/adjacent = y/x

use pythagora's theorem to get y

y² + x² = 10²

remember we are looking for how fast the angle is changing when x = 6ft

y² + 6² = 100

y² + 36 = 100

y² = 100 - 36

y²  = 64

y = √64

y = 8

Rates of change

dx/dt = 1.5 ft/s

we are asked to find

d∅/dt when x = 6 ft

<em>tan ∅ = y/x</em>

find the derivative using the quotient rule

sec²∅ d∅/dt = (x dy/dt - y dx/dt) / x²

sec²∅ d∅/dt = (6 dy/dt - 8 dx/dt )/ 36

recall sec²∅ = (sec ∅)²

sec ∅ = 1/cos ∅

since cos ∅ = adjacent/ hypotenuse

sec ∅ = hypotenuse/adjacent

sec ∅ = 10/6

(sec ∅)² = 100/36

To get dy/dt

y² + x² = 10² dh/dt

find the derivative

2y dy/dt + 2x dx/dt = 100 (0)

note the length of ladder does not change

2 (8) dy/dt + 2 (6) 1.5 = 0

16 dy/dt + 18 = 0

dy/dt = -18/16

dy/dt = - 9/8

dy/dt = - 1.125

Therefore,

sec²∅ d∅/dt = (6 dy/dt - 8 dx/dt )/ 36

100/36 d∅/dt = (6 (-1.125) - 8 (1.5) )/36

100/36 d∅/dt = ( -6.75 - 12)/36

100/36 d∅/dt = - 18.75/36

100/36 d∅/dt = - 0.52083333333

2.77777777778  d∅/dt = -0.52083333333

divide both sides by 2.77777777778

d∅/dt =  -0.52083333333/2.77777777778

d∅/dt = - 0.18101999999  rad/s

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If we talk about teh atmospheric pressure, the density of air goes on decreasing as we go up and up. o we cannot say that it is directly depends only on the depth of air, it also depends on the changing density of air.

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Suppose a rocket ship in deep space moves with constant acceleration equal to 9.80 m/s2, which gives the illusion of normal grav
DochEvi [55]

Answer:

a) 3673469.39 seconds

b) 6.61×10¹⁴ m

Explanation:

t = Time taken

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s = Displacement

a = Acceleration due to gravity = 9.8 m/s²

Equation of motion

v=u+at\\\Rightarrow 0.12\times 3\times 10^8=0+9.8t\\\Rightarrow t=\frac{0.12\times 3\times 10^8}{9.8}=3673469.39\ s

Time taken to reach 12% of light speed is 3673469.39 seconds

v^2-u^2=2as\\\Rightarrow s=\frac{v^2-u^2}{2a}\\\Rightarrow s=\frac{(0.12\times 3\times 10^8)^2-0^2}{2\times 9.8}\\\Rightarrow s=6.61 \times 10^{14}\ m

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3 years ago
Using this formula a = F/m What acceleration results from exerting a 125N force on a 0.65kg
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Answer:

Acceleration = 192.3 m/s² (Approx.)

Explanation:

Given:

Force = 125 N

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Acceleration = Force / Mas

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A vertical spring has a spring constant of 2900 N/m. The spring is compressed 80 cm and a 8 kg spider is placed on the spring. T
Serga [27]

Answer:

a)  k_{e} = 928 J , b)U = -62.7 J , c) K = 0 , d) Y = 11.0367 m,  e)  v = 15.23 m / s  

Explanation:

To solve this exercise we will use the concepts of mechanical energy.

a) The elastic potential energy is

      k_{e} = ½ k x²

      k_{e} = ½ 2900 0.80²

      k_{e} = 928 J

b) place the origin at the point of the uncompressed spring, the spider's potential energy

     U = m h and

     U = 8 9.8 (-0.80)

     U = -62.7 J

c) Before releasing the spring the spider is still, so its true speed and therefore the kinetic energy also

      K = ½ m v²

      K = 0

d) write the energy at two points, maximum compression and maximum height

     Em₀ = ke = ½ m x²

     E_{mf} = mg y

     Emo = E_{mf}

     ½ k x² = m g y

     y = ½ k x² / m g

     y = ½ 2900 0.8² / (8 9.8)

     y = 11.8367 m

As zero was placed for the spring without stretching the height from that reference is

     Y = y- 0.80

     Y = 11.8367 -0.80

     Y = 11.0367 m

Bonus

Energy for maximum compression and uncompressed spring

     Emo = ½ k x² = 928 J

     E_{mf}= ½ m v²

     Emo = E_{mf}

     Emo = ½ m v²

      v =√ 2Emo / m

     v = √ (2 928/8)

     v = 15.23 m / s

8 0
3 years ago
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