A 20 kg object is acted on by a conservative force given by F = -3.0 x - 5.0 x 2, 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 = 3.0 m?
______ J
(b) If the object has a velocity of 2.0 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 -7.0 J when the object is at x = 0?
(a) _______ J
(b) _______ m/s
To find the potential energy associated with the force when the object is at a specific position, we can use the formula for potential energy:
Potential Energy (PE) = -∫ F(x) dx
where F(x) is the force as a function of position.
(a) To find the potential energy at x = 3.0 m, we need to integrate the force function from x = 0 to x = 3.0 m.
PE = -∫ [(-3.0x - 5.0)(2)] dx
Integrating this expression gives:
PE = -[(-3.0/2)x^2 - 5.0x] evaluated from 0 to 3.0 m
PE = -[(-3.0/2)(3.0)^2 - 5.0(3.0)] + [(-3.0/2)(0)^2 - 5.0(0)]
PE = -[(-3.0/2)(9.0) - 15.0]
PE = -(13.5 - 15.0)
PE = -(-1.5)
PE = 1.5 J
Therefore, the potential energy of the system associated with the force when the object is at x = 3.0 m is 1.5 J.
(b) To find the speed of the object when it passes through the origin, we can use the principle of conservation of mechanical energy. At any position, the total mechanical energy of the system is the sum of the kinetic energy and potential energy.
Total Mechanical Energy (E) = Kinetic Energy (KE) + Potential Energy (PE)
When the object is at x = 6.0 m, its mechanical energy is equal to the sum of its kinetic and potential energies at that position:
E = KE + PE
Since the force is conservative, the total mechanical energy remains constant.
When the object passes through the origin, we know that its position is x = 0. At this point, the potential energy is given as -7.0 J.
E = KE + PE
-7.0 J = KE + 0
We can rearrange the equation to solve for the kinetic energy:
KE = -7.0 J
To find the velocity, we can use the formula for kinetic energy:
KE = (1/2)mv^2
where m is the mass of the object and v is the velocity.
Since the mass (m) of the object is not given, we cannot determine the exact speed when it passes through the origin. We would need the mass of the object to calculate the velocity.
(c) If the potential energy is taken to be -7.0 J when the object is at x = 0, the potential energy expression changes.
(a) To find the potential energy at x = 3.0 m:
PE = -∫ [(-3.0x - 5.0)(2)] dx
PE = -[(-3.0/2)x^2 - 5.0x] evaluated from 0 to 3.0 m
PE = -[(-3.0/2)(3.0)^2 - 5.0(3.0)] + [(-3.0/2)(0)^2 - 5.0(0)]
PE = -[(-3.0/2)(9.0) - 15.0]
PE = -(13.5 - 15.0)
PE = -(-1.5)
PE = 1.5 J
The potential energy of the system associated with the force when the object is at x = 3.0 m is still 1.5 J, regardless of the chosen reference point for potential energy.
(b) Without the mass of the object, we cannot determine the exact speed when it passes through the origin. Therefore, the answer remains unknown.