A 20 kg object is acted on by a conservative force given by...

A $20 \mathrm{~kg}$ 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 $x=2.0 \mathrm{~m} ?(\mathrm{~b})$ If the object has a velocity of $4.0 \mathrm{~m} / \mathrm{s}$ in the negative direction of the $x$ axis when it is at $x=5.0 \mathrm{~m}$, 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 $-8.0 \mathrm{~J}$ when the object is at $x=0$ ?

Key Concepts

Reference Point for Potential Energy

The zero level of potential energy is arbitrary and can be chosen based on convenience or the specifics of a problem. Changing the reference point alters the numerical value of the potential energy but does not affect the physical predictions since only differences in potential energy are relevant. This concept is important when comparing energy values calculated with different potential energy baselines.

Energy Conservation

The principle of energy conservation states that the total mechanical energy of a system, which is the sum of its kinetic and potential energies, remains constant if only conservative forces perform work. This principle allows the analysis of motion by equating energy at different positions, making it possible to solve for quantities like speed when an object moves between points.

Conservative Forces

A conservative force is one for which the work done in moving an object between two points is independent of the trajectory taken. This allows the definition of a potential energy function such that the force can be derived as the negative gradient of the potential energy. As a result, energy methods can be applied to analyze the motion of objects subject to conservative forces.

Potential Energy

Potential energy is the stored energy associated with the position or configuration of an object within a force field. In the context of conservative forces, the potential energy function can be obtained by integrating the force with respect to position and is defined up to an arbitrary constant. This constant can be chosen based on convenience, such as setting the potential energy to zero at a specific reference point.

Transcript

00:02 Here we know that from equation 8, 6, we can find the potential energy in terms of x.

00:09 This would be the negative integral from 0 to x of negative 3 .0x minus 5 .0x squared.

00:17 And we can say that the potential energy in terms of x would be equal to 3x squared over 2.

00:24 And then this would be plus 5x cubed over 3.

00:31 We can then say for part a using the formula, the u at x equals 2 would be equal to 3 times 2 squared divided by 2 plus 5 times 2 cubed divided by 3, and the potential energy at x equals 2 is going to be equal to approximately 19 joules.

00:51 We can then say for part b, we know that when the speed is equal to 4, four meters per second, the mechanical energy is going to be equal to one half mv squared plus the potential energy at five.

01:05 So we can then say that the speed, or rather this must equal the energy at the origin.

01:10 So we can say that for part b, this would be the total energy, total mechanical energy.

01:21 And we can then say that four part b, one half mv squared plus the potential energy, and we can at 5 will be equal to 1 half times the initial velocity plus the potential energy at 0.

01:36 So we know that here the potential energy at zero is going to be zero.

01:41 So the initial velocity would simply be equal to the square root of the velocity squared plus 2 times the potential energy at 5 divided by the mass.

01:54 And then we can then say this is going to be equal to 4 squared plus 2, and then the potential energy at 5 is 246 joules...

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