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
How many days are in 345000 minutes? (Set up by dimensional analysis)​
Komok [63]
239.583 days is the answer
3 0
2 years ago
Discribe the law of reflection
aniked [119]

Answer:

the angle of reflection equals the angle of incidence—θr = θi. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.

Explanation:

A microscope uses a mirror to reflect light to the specimen under the microscope. ... An astronomical reflecting telescope uses a large parabolic mirror to gather dim light from distant stars. A plane mirror is used to reflect the image to the eyepiece.

8 0
3 years ago
A parallel-plate capacitor is disconnected from a battery, and the plates are pulled a small distance farther apart. Does Q incr
kolezko [41]

Answer:

Q stay the same

Explanation:

Charging of capacitor is done by battery . If battery is disconnected , charging will stop . There will not be any discharging as plates are separate . So pulling the plates apart will not affect the charge lying on the capacitor . It will decrease its capacity and increase its potential , keeping its charge constant.

4 0
3 years ago
A toroid has a square cross section with the length of an edge equal to the radius of the inner surface. The ratio of the magnit
Tasya [4]

Answer:

2

Explanation:

7 0
3 years ago
Read 2 more answers
In a double-slit experiment, the slits are illuminated by a monochromatic, coherent light source having a wavelength of 609 nm.
Ray Of Light [21]

Answer:

\Delta x = 3.65 \mu m

Explanation:

As we know that the sixth order maximum will have path difference given as

\Delta x = N\lambda

here we know that

N = order of maximum

\lambda = 609 nm

now we have

N = 6

so we know that

\Delta x = 6(609 nm)

\Delta x = 3.65 \mu m

6 0
3 years ago
Other questions:
  • Screws and wedges are modified blank
    15·1 answer
  • Podocytes are also called<br> 1 poin
    5·1 answer
  • A changing magnetic field can produce an electric current. True or False?
    15·1 answer
  • If a steel containing 1.88 wt%C is cooled relatively slowly to room temperature, what is the expected weight fraction of pearlit
    5·1 answer
  • A loop of wire carries a current. the resulting magnetic field __________.
    14·1 answer
  • Is the volcano mount tabors dangerous
    9·1 answer
  • A 12,500 N alien UFO is hovering about the surface of Earth. At time , its position can be given as () = ((0.24 m/s^3)^3 + 25 m)
    10·1 answer
  • Why aren't homes wired in series? A. Because it's too expensive. B. Parallel circuits are too complex for home wiring. O C. Beca
    10·2 answers
  • What do your joints do?
    12·2 answers
  • Select the correct answer.<br> Which type of energy is thermal energy a form of
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!