A 15 kg object is acted on by a conservative force given by F = -1.7x - 4.7x2, w

Question

A 15 kg object is acted on by a conservative force given by F = -1.7x - 4.7x2, 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 = 2.3 m? (b) If the object has a velocity of 2.9 m/s in the negative direction of the x axis when it is at x = 6.3 m, what is its speed when it passes through the origin? What are the answers to (c) (a) and (d) (b) if the potential energy of the system is taken to be -12 J when the object is at x = 0?

Explanation / Answer

The mass of the object m =15 kg
The conservative force F = (-1.7x - 4.7x^2) N
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a) the potential energy u =-integral(F dx),limits x_i =0 m to x_f =2.3 m
U =-integral( (-1.7x - 4.7x^2) dx),limits x_i =0 m to x_f =2.3 m
U =-[-1.7 x^2/2-4.7x^3/3],limits x_i =0 m to x_f =2.3 m
U =23.55 J
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b)
From the law of conservation of energy,
The change in kinetic energy = Potential energy
K_f-K_i =integral (F dx),limits x_i =6.3 m to x_f =0 m
=[-1.7 x^2/2-4.7x^3/3],limits x_i =6.3 m to x_f =0 m
=425.47 J 1/2(m)[(v_f)^2 -(v_i)^2] =425.47 J where v_i =2.9 m/s v_f =8.07 m/s ------------------------------------------------------------------------------------------- c) at x =2.3 m potential energy U(x) =1.7/2(x^2)+4.7/3(x^3)-12J =23.55 J-12 J =11.55 J ------------------------------------------------------------------------------------------- d) at x =6.3 m potential energy U(x) =1.7/2(x^2)+4.7/3(x^3)-12J =425.47 J-12 J=413.47 J velocity v_f =7.97 m/s

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