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Tom [10]
3 years ago
11

A particle of mass m moves under the influence of a force given by F = (−kx + kx3/α2) where k and α are positive constants. a) F

ind the potential energy function U(x) (Take U(x=0) = 0). b) Find any equilibrium points and determine if they are stable or unstable. c) Sketch the potential as a function of position. d)What is the maximum total energy, Emax a particle may have if its motion is to remain bounded? e) For the case of bounded motion with E
Physics
1 answer:
Romashka-Z-Leto [24]3 years ago
4 0

Answer:

Explanation:

a ) F = (-kx + kx³/a²)

intensity of field

I = F / m

=  (-kx + kx³/a²) / m

If U be potential function

- dU / dx =  (-kx + kx³/a²) / m

U(x)  = ∫  (kx - kx³/a²) / m dx

= k/m ( x²/2 - x⁴/4a²)

b )

For equilibrium points , U is either maximum or minimum .

dU / dx = x - 4x³/4a² = 0

x = ± a.

dU / dx = x - x³/a²

Again differentiating

d²U / dx² = 1 - 3x² / a²

Put the value of x = ± a.

we get

d²U / dx²  = -2 ( negative )

So at x = ± a , potential energy U is maximum.

c )

U =  k/m ( x²/2 - x⁴/4a²)

When x =0 , U = 0

When x=  ± a.

U is maximum

So the shape of the U-x curve is like a bowl centered at x = 0

d ) Maximum potential energy

put x = a or -a in

U(max)  =  k/m ( x²/2 - x⁴/4a²)

= k/m ( a² / 2 - a⁴/4a²)

= k/m ( a² / 2 - a²/4)

a²k / 4m

This is the maximum total energy where kinetic energy is zero.

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Zarrin [17]

Answer:

A. 69.9m

Explanation:

Given parameters:

Initial velocity = 10.5m/s

Final velocity  = 21.7m/s

Time  = 4.34s

Unknown:

Distance traveled = ?

Solution:

Let us first find the acceleration of the car;

  Acceleration  = \frac{v - u}{t}

  v is final velocity

   u is initial velocity

   t is the time

     Acceleration  = \frac{21.7 - 10.5}{4.34}   = 2.58m/s²

Distance traveled;

     V² = U² + 2aS

    21.7² = 10.5² + 2 x 2.58 x S

   360.64 = 2 x 2.58 x S

     S = 69.9m

3 0
3 years ago
Please solve this question ​
lesantik [10]

Answer:

88200 Pa

it is because

height =9m

density=1000kg/m(cube)

gravity = 9.8m/s(square)

now,

P=d×g×h

= 1000×9.8×9

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8 0
2 years ago
Using energy considerations, calculate the average force (in N) a 67.0 kg sprinter exerts backward on the track to accelerate fr
Illusion [34]

Answer:

F_{sprinter}=110.4N

Explanation:

Given data

Mass m=67.0 kg

Final Speed vf=8.00 m/s

Initial Speed vi=2.00 m/s

Distance d=25.0 m

Force F=30.0 N

From work-energy theorem we know that the work done equals the change in kinetic energy

W=ΔK=Kf-Ki=1/2mvf²-1/2mvi²

And

W=F_{total}.d

So

W=1/2mv_{f}^2-1/2mv_{i}^2\\F_{total}=\frac{1/2mv_{f}^2-1/2mv_{i}^2}{d} \\F_{total}=\frac{1/2(67.0kg)(8.00m/s)^2-1/2(67.0kg)(2.00m/s)^2}{25.0m} \\F_{total}=80.4N

and we know that the force the sprinter exerted Fsprinter the force of the headwind Fwind=30.0N

So

F_{sprinter}=F_{total}+F_{wind}\\F_{sprinter}=80.4N+30N\\F_{sprinter}=110.4N  

7 0
3 years ago
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A standard 1 kilogram weight is a cylinder 54.0 mm in height and 55.0 mm in diameter. what is the density of the material
denis-greek [22]

The radius of the cylinder is equal to half the diameter:

r=\frac{d}{2}=\frac{55.0 mm}{2}=27.5 mm

The volume of the cylinder is given by:

V=\pi r^2 h=\pi (27.5 mm)^2 (54.0 mm)=1.28 \cdot 10^5 mm^3

where h is the heigth of the cylinder. Converting into meters,

V=1.28 \cdot 10^{-4} m^3

And the density of the material will be given by the ratio between the mass and the volume:

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A potential energy function is given by U(x)=(3.00J)x+(1.00J/m2)x3. What is the force function F(x) (in newtons) that is associa
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Answer:

F(x)=-3 N - (3N)x^2

Explanation:

The force is defined as the negative of the derivative of the potential energy:

F=-\frac{dU}{dx}

If we use the potential energy function given in this problem:

U(x)=3.00 x + 1.00 x^3

and we calculate the force, we get:

F(x)=-\frac{d}{dx}(3x+x^3)=-3-3x^2

So, the force is

F(x)=-3 N - (3N)x^2

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3 years ago
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