A 20 kg object is acted on by a conservative force given by F = 3.0 x 5.0 x 2 ,

Question

A 20 kg object is acted on by a conservative force given by F = 3.0x 5.0x2, with F in newtons and x in meters. Take the potential energy associated with the force to be zero when the object is at x = 0.

(a) What is the potential energy of the system associated with the force when the object is at x = 6.0 m?

(b) If the object has a velocity of 3.0 m/s in the negative direction of the x axis when it is at x = 6.0 m, what is its speed when it passes through the origin?

(c) What are the answers to (a) and (b) if the potential energy of the system is taken to be 5.0 J when the object is at x = 0?

(a)____ J (b) ____ m/s A 20 kg object is acted on by a conservative force given by F--3.0x 5.0x2, with F in newtons and x in meters. Take the potential energy associated with the force to be zero when the object is atx O (a) What is the potential energy of the system associated with the force when the object is at x = 6.0 m? (b) If the object has a velocity of 3.0 m/s in the negative direction of the x axis when it is at x = 6.0 m, what is its speed when it passes through the origin? m/s (c) What are the answers to (a) and (b) if the potential energy of the system is taken to be-5.0 J when the object is at x = 0? m/s

Explanation / Answer

a) Fc=-dU/dx
So dU=-f*dx
Integrating with respect to x
3x²/2 + 5x³/3 put limit 0 to 6
= 414 J


b) F= dp/dt
dp= F dt
or
dp= F dx dt/dx
dp dx/dt= F dx
m v²= F dx
Integrating
mv³/2 = -3x²/2 - 5x³/3
putting limit from x = 6 to x=0
mv³= 828
solving and putting value of m and taking root
v= 3.42 m/s

c) Using same function--
dU= -F dx
Integrating U from U= -5 J to U = U'
and Integrating x from x = 0 to x = x
we get:
U + 5J= 3x²/2 + 5x³/3
now put x = 6
U'+ 5J = 414
U'= 409 J

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