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
Which of the following is an example of an insulator?
charle [14.2K]

Answer:

Hey mate

Answer is glass

7 0
2 years ago
Read 2 more answers
How does the government shape technology?
Marat540 [252]

Answer:

Society is a group of people that share similar values and beliefs.

The development of technology is affected by society and its changing values, politics, and economics

Explanation:

8 0
3 years ago
A suspended object A is attracted to a charged object B, can one conclude that A is charged? Explain
irinina [24]

Explanation:

In my view, when the Object A is attracted to a Charged object B. Object B should be Negatively or Positively charged. So Object B should be the Opposite charged according to the Object B

Example =

If Object B is Negatively Charged, the Object A should be Positively Charged

If the Object B is Positively Charged, the Object A should be Negatively Charged

Sometimes it can Mix as a Neutral as well

Hope this Helps

5 0
3 years ago
When a sacred item or symbol is removed from its special place or is duplicated in mass quantities, then it becomes profane as a
Galina-37 [17]

Answer:

It becomes profane as a result of desacralization

Explanation:

Because desacralization means when a dedicated religious structure is no longer used for its intended purpose both rather used for another purpose other than the original purpose

7 0
3 years ago
A girl is floating in a freshwater lake with her head just above the water. If she weighs 610 N, what is the volume of the subme
Elden [556K]

Answer:

The volume of the submerged part of her body is 0.0622m^{3}

Explanation:

Let's define the buoyant force acting on a submerged object.

In a submerged object acts a buoyant force which can be calculated as :

B=ρ.V.g

Where ''B'' is the buoyant force

Where ''ρ'' is the density of the fluid

Where ''V'' is the submerged volume of the object

Where ''g'' is the acceleration due to gravity

Because the girl is floating we can state that the weight of the girl is equal to the buoyant force.

We can write :

W_{girl}=B (I)

Where ''W'' is weight

⇒ If we consider ρ = 1000\frac{kg}{m^{3}} (water density) and g=9.81\frac{m}{s^{2}} and replacing this values in the equation (I) ⇒

B=W_{girl}

B=610N

ρ.V.g = 610N

1000\frac{kg}{m^{3}}.V.(9.81\frac{m}{s^{2}})=610N (II)

The force unit ''N'' (Newton) is defined as

N=kg.\frac{m}{s^{2}}

Using this in the equation (II) :

(9810\frac{N}{m^{3}}).V =610N

V=\frac{610N}{9810\frac{N}{m^{3}}}

V=0.0622m^{3}

We find that the volume of the submerged part of her body is 0.0622m^{3}

8 0
3 years ago
Other questions:
  • to what height will a 250g soccer ball rise to if it is kicked directly upwards at 8 meters per second​
    6·2 answers
  • 38 Points + Brainlyest!!!
    8·2 answers
  • How do you know that potassium an alkali metal is highly reactive
    5·1 answer
  • For a given Prandtl-Meyer expansion, the upstream Mach number is 3 and the pressure ratio across the wave is P2/P1 = 0.4. Calcul
    6·1 answer
  • How you could use a graduated cylinder to find the volume of a small rock
    6·1 answer
  • A circuit has two 4Ω resistors in series with each other. What is the total resistance?
    11·1 answer
  • A physics major is cooking breakfast when he notices that the frictional force between the steel spatula and the Teflon frying p
    12·1 answer
  • A common, though incorrect, statement is, "The Moon orbits the Earth." That creates an image of the Moon?s orbit that looks like
    7·1 answer
  • It took 500 N of force to push a car 4 meters. How much work was done?
    6·1 answer
  • So I kicked my leg in the air trying to do something but I slipped cause I had socks on. I hit the ground hard with my left hand
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!