1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tia_tia [17]
3 years ago
8

Find analytically the velocity of the object at the end point of the inclined plane for a certain angle Ө

Physics
1 answer:
goldfiish [28.3K]3 years ago
8 0

I don't know if there is other given information that's missing here, so I'll try to fill in the gaps as best I can.

Let <em>m</em> be the mass of the object and <em>v</em>₀ its initial velocity at some distance <em>x</em> up the plane. Then the velocity <em>v</em> of the object at the bottom of the plane can be determined via the equation

<em>v</em>² - <em>v</em>₀² = 2 <em>a</em> <em>x</em>

where <em>a</em> is the acceleration.

At any point during its motion down the plane, the net force acting on the object points in the same direction. If friction is negligible, the only forces acting on the object are due to its weight (magnitude <em>w</em>) and the normal force (mag. <em>n</em>); if there is friction, let <em>f</em> denote its magnitude and let <em>µ</em> denote the coefficient of kinetic friction.

Recall Newton's second law,

∑ <em>F</em> = <em>m</em> <em>a</em>

where the symbols in boldface are vectors.

Split up the forces into their horizontal and vertical components. Then by Newton's second law,

• net horizontal force:

∑ <em>F</em> = <em>n</em> cos(<em>θ</em> + 90º) = <em>m</em> <em>a</em> cos(<em>θ</em> + 180º)

→  - <em>n</em> sin(<em>θ</em>) = - <em>m</em> <em>a</em> cos(<em>θ</em>)

→  <em>n</em> sin(<em>θ</em>) = <em>m</em> <em>a</em> cos(<em>θ</em>) ……… [1]

• net vertical force:

∑ <em>F</em> = <em>n</em> sin(<em>θ</em> + 90º) - <em>w</em> = <em>m</em> <em>a</em> sin(<em>θ</em> + 180º)

→   <em>n</em> cos(<em>θ</em>) - <em>m</em> <em>g</em> = - <em>m</em> <em>a</em> sin(<em>θ</em>)

→   <em>n</em> cos(<em>θ</em>) = <em>m</em> (<em>g</em> - <em>a</em> sin(<em>θ</em>)) ……… [2]

where in both equations, <em>a</em> is the magnitude of acceleration, <em>g</em> = 9.80 m/s², and friction is ignored.

Then by multiplying [1] by cos(<em>θ</em>) and [2] by sin(<em>θ</em>), we have

<em>n</em> sin(<em>θ</em>) cos(<em>θ</em>) = <em>m</em> <em>a</em> cos²(<em>θ</em>)

<em>n</em> cos(<em>θ</em>) sin(<em>θ</em>) = <em>m</em> (<em>g</em> sin(<em>θ</em>) - <em>a</em> sin²(<em>θ</em>))

<em>m</em> <em>a</em> cos²(<em>θ</em>) = <em>m</em> (<em>g</em> sin(<em>θ</em>) - <em>a</em> sin²(<em>θ</em>))

<em>a</em> cos²(<em>θ</em>) + <em>a</em> sin²(<em>θ</em>) = <em>g</em> sin(<em>θ</em>)

<em>a</em> = <em>g</em> sin(<em>θ</em>)

and so the object attains a velocity of

<em>v</em> = √(<em>v</em>₀² + 2 <em>g</em> <em>x</em> sin(<em>θ</em>))

If there is friction to consider, then <em>f</em> = <em>µ</em> <em>n</em>, and Newton's second law instead gives

• net horizontal force:

∑ <em>F</em> = <em>n</em> cos(<em>θ</em> + 90º) + <em>f</em> cos(<em>θ</em>) = <em>m</em> <em>a</em> cos(<em>θ</em> + 180º)

→   - <em>n</em> sin(<em>θ</em>) + <em>µ</em> <em>n</em> cos(<em>θ</em>) = - <em>m</em> <em>a</em> cos(<em>θ</em>)

→   <em>n</em> sin(<em>θ</em>) - <em>µ</em> <em>n</em> cos(<em>θ</em>) = <em>m</em> <em>a</em> cos(<em>θ</em>) ……… [3]

• net vertical force:

∑ <em>F</em> = <em>n</em> sin(<em>θ</em> + 90º) + <em>f</em> sin(<em>θ</em>) - <em>w</em> = <em>m</em> <em>a</em> sin(<em>θ</em> + 180º)

→   <em>n</em> cos(<em>θ</em>) + <em>µ</em> <em>n</em> sin(<em>θ</em>) - <em>m</em> <em>g</em> = - <em>m</em> <em>a</em> sin(<em>θ</em>)

→   <em>n</em> cos(<em>θ</em>) + <em>µ</em> <em>n</em> sin(<em>θ</em>) = <em>m</em> <em>g</em> - <em>m</em> <em>a</em> sin(<em>θ</em>) ……… [4]

Then multiply [3] by cos(<em>θ</em>) and [4] by sin(<em>θ</em>) to get

- <em>n</em> sin(<em>θ</em>) cos(<em>θ</em>) + <em>µ</em> <em>n</em> cos²(<em>θ</em>) = - <em>m</em> <em>a</em> cos²(<em>θ</em>)

<em>n</em> cos(<em>θ</em>) sin(<em>θ</em>) + <em>µ</em> <em>n</em> sin²(<em>θ</em>) = <em>m</em> <em>g</em> sin(<em>θ</em>) - <em>m</em> <em>a</em> sin²(<em>θ</em>)

and adding these together gives

<em>µ</em> <em>n</em> (cos²(<em>θ</em>) + sin²(<em>θ</em>)) = <em>m</em> <em>g</em> sin(<em>θ</em>) - <em>m</em> <em>a</em> (cos²(<em>θ</em>) + sin²(<em>θ</em>))

<em>µ</em> <em>n</em> = <em>m</em> <em>g</em> sin(<em>θ</em>) - <em>m</em> <em>a</em>

<em>m a</em> = <em>m</em> <em>g</em> sin(<em>θ</em>) - <em>µ</em> <em>n</em>

<em>m a</em> = <em>m</em> <em>g</em> sin(<em>θ</em>) - <em>µ</em> <em>m</em> <em>g</em> cos (<em>θ</em>)

<em>a</em> = <em>g</em> (sin(<em>θ</em>) - <em>µ</em> cos (<em>θ</em>))

and so the object would instead attain a velocity of

<em>v</em> = √(<em>v</em>₀² + 2 <em>g</em> <em>x</em> (sin(<em>θ</em>) - <em>µ</em> cos (<em>θ</em>)))

You might be interested in
What happens to light when it changes speed?
Brut [27]
Hey there!

When light changes speed, it REFRACTS.
Your answer is going to be option C.

Hope this helps you.
Have a great day!
5 0
3 years ago
Which energy transformation occurs in the core of a nuclear reactor?
romanna [79]
The correct answer among the choices given is option B. The energy transformation that occurs in the core of a nuclear reactor is from nuclear energy to thermal energy. In a power plant nuclear fission which involves nuclear energy to heat up water around it. This part is  the core of the process.
5 0
3 years ago
Tonya is thinking about the topic presented in the text, "Do opposites really attract?" Which of her thoughts is an example of c
tigry1 [53]

tanya is dumb  j j j j j j j j j jj j j j

6 0
3 years ago
What is the all time speed record for completing the iditarod?.
shtirl [24]
<h3>Question:</h3>

•What is the all time speed record for completing the iditarod?

Answer:

•In 2016, Dallas broke his own record, finishing in 8 days, 11 hours, 20 minutes and 16 seconds. In 2017, Mitch Seavey broke all previous records by finishing in 8 days, 3 hours, 40 minutes and 13 seconds, which currently stands as the fastest winning time for the Iditarod.

Explanation:

#Let's Study

#I Hope It's Help

#Keep On Learning

#Carry On Learning

3 0
2 years ago
An 87.6 g lead ball is dropped from rest from a height of 7.00 m. The collision between the ball and the ground is totally inela
bija089 [108]

Taking specific heat of lead as 0.128 J/gK = c

We have energy of ball at 7.00 meter height = mgh = 87.6*10^{-3}*9.81*7

When leads gets heated by a temperature ΔT energy needed = mcΔT

                                                                      = 87.6*10^{-3}*0.128*10^3ΔT

Comparing both the equations

                      87.6*10^{-3}*9.81*7 = 87.6*10^{-3}*0.128*10^3ΔT

                        ΔT = 0.536 K

                        Change in temperature same in degree and kelvin scale

                                      So ΔT = 0.536 ^0C

7 0
2 years ago
Other questions:
  • A rocket blasts off vertically from rest on the launch pad with a constant upward acceleration of 2.30 m/s2. At 20.0 s after bla
    15·1 answer
  • Suppose that on a hot and sticky afternoon in the spring, a tornado passes over the high school. If the air pressure in the lab
    14·1 answer
  • In fair weather, the ground may become charged such that there is an electric field just above the surface of the Earth, pointin
    7·1 answer
  • A certain part of a flat screen TV has a thickness of 150 nanometers. How<br> many meters is this?
    13·1 answer
  • The light bulb in your room is not very bright. You want to use a concave mirror to direct its illumination into a tight, parall
    13·1 answer
  • Four charges are at the corners of a square centered at the origin as follows q at a a 2q at a a 3q at a a and 6q at a a A fifth
    13·1 answer
  • A system gains 767 kJ of heat, resulting in a change in internal energy of the system equal to +151 kJ. How much work is done?
    10·1 answer
  • Student one used bowling ball A in a bowling game against Student 2, who used bowling ball B. Use Newton’s Three Laws of Motion
    13·1 answer
  • lodine-131 has a half life of about 8 days. If you had 60 grams of , how much would remain after 24 days?
    9·1 answer
  • After a meal.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!