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Chapter 6: Problem 36

The potential energy of a certain particle is given by \(U=10 x^{2}+35 z^{3}\). Find the force vector exerted on the particle.

Short Answer

Expert verified
Answer: The force vector exerted on the particle is \(F = \langle -20x, -105z^2 \rangle\).

Step by step solution

01

Calculate the partial derivative of U with respect to x

To find the partial derivative of U with respect to x, we differentiate U with respect to x, treating z as a constant. \[\frac{\partial U}{\partial x} = 20x\]
02

Calculate the partial derivative of U with respect to z

To find the partial derivative of U with respect to z, we differentiate U with respect to z, treating x as a constant. \[\frac{\partial U}{\partial z} = 105z^2\]
03

Represent the partial derivatives as a force vector

We will represent the force vector (F) as \[F = \langle F_x, F_z \rangle\]
04

Calculate the components of the force vector

To calculate the components of the force vector, we apply the negative gradient of the potential energy function. So, we will have \[F_x = -\frac{\partial U}{\partial x} = -20x\] and \[F_z = -\frac{\partial U}{\partial z} = -105z^2\]
05

Write the force vector

Now that we have the components of the force vector, we can write it as: \[F = \langle -20x, -105z^2 \rangle\] Therefore, the force vector exerted on the particle is \(F = \langle -20x, -105z^2 \rangle\).

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Most popular questions from this chapter

The molecular bonding in a diatomic molecule such as the nitrogen \(\left(\mathrm{N}_{2}\right)\) molecule can be modeled by the Lennard Jones potential, which has the form $$ U(x)=4 U_{0}\left(\left(\frac{x_{0}}{x}\right)^{12}-\left(\frac{x_{0}}{x}\right)^{6}\right) $$ where \(x\) is the separation distance between the two nuclei and \(x_{0}\), and \(U_{0}\) are constants. Determine, in terms of these constants, the following: a) the corresponding force function; b) the equilibrium separation \(x_{0}\), which is the value of \(x\) for which the two atoms experience zero force from each other; and c) the nature of the interaction (repulsive or attractive) for separations larger and smaller than \(x_{0}\).

Which of the following is not a unit of energy? a) newton-meter b) joule c) kilowatt-hour d) \(\operatorname{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\) e) all of the above

A variable force acting on a 0.100 - \(\mathrm{kg}\) particle moving in the \(x y\) -plane is given by \(F(x, y)=\left(x^{2} \hat{x}+y^{2} \hat{y}\right) \mathrm{N},\) where \(x\) and \(y\) are in meters. Suppose that due to this force, the particle moves from the origin, \(O\), to point \(S\), with coordinates \((10.0 \mathrm{~m},\) \(10.0 \mathrm{~m}\) ). The coordinates of points \(P\) and \(Q\) are \((0 \mathrm{~m}, 10.0 \mathrm{~m})\) and \((10.0 \mathrm{~m}, 0 \mathrm{~m})\) respectively. Determine the work performed by the force as the particle moves along each of the following paths: a) OPS b) OQS c) OS d) \(O P S Q O\) e) \(O Q S P O\)

What is the gravitational potential energy of a \(2.0-\mathrm{kg}\) book \(1.5 \mathrm{~m}\) above the floor?

A snowboarder of mass \(70.1 \mathrm{~kg}\) (including gear and clothing), starting with a speed of \(5.1 \mathrm{~m} / \mathrm{s}\), slides down a slope at an angle \(\theta=37.1^{\circ}\) with the horizontal. The coefficient of kinetic friction is \(0.116 .\) What is the net work done on the snowboarder in the first 5.72 s of descent?
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