8.4 Potential energy diagrams and stability  (Page 4/7)

A mysterious constant force of 10 N acts horizontally on everything. The direction of the force is found to be always pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distance x from the wall, assuming the potential energy at the wall to be zero.

A single force $F\left(x\right)=\mathrm{-4.0}x$ (in newtons) acts on a 1.0-kg body. When $x=3.5\text{m,}$ the speed of the body is 4.0 m/s. What is its speed at $x=2.0\text{m?}$

A particle of mass 4.0 kg is constrained to move along the x -axis under a single force $F\left(x\right)=\text{−}c{x}^3,$ where $c=8.0{\text{N/m}}^3.$ The particle’s speed at A , where $x_A=1.0\text{m,}$ is 6.0 m/s. What is its speed at B , where $x_B=\mathrm{-2.0}\text{m?}$

The force on a particle of mass 2.0 kg varies with position according to $F\left(x\right)=\mathrm{-3.0}{x}^2$ ( x in meters, F ( x ) in newtons). The particle’s velocity at $x=2.0\text{m}$ is 5.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) $x=4.0\text{m}$ as the reference point. (c) Find the particle’s velocity at $x=1.0\text{m}.$ Do this part of the problem for each reference point.

A 4.0-kg particle moving along the x -axis is acted upon by the force whose functional form appears below. The velocity of the particle at $x=0$ is $v=6.0\text{m/s}.$ Find the particle’s speed at $x=\left(\text{a}\right)2.0\text{m},\left(\text{b}\right)4.0\text{m},\left(\text{c}\right)10.0\text{m},\left(\text{d}\right)$ Does the particle turn around at some point and head back toward the origin? (e) Repeat part (d) if $v=2.0\text{m/s}\text{at}x=0.$

A particle of mass 0.50 kg moves along the x -axis with a potential energy whose dependence on x is shown below. (a) What is the force on the particle at $x=2.0,5.0,8.0,\text{and}$ 12 m? (b) If the total mechanical energy E of the particle is −6.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if $E=2.0\text{J?}$ (d) If $E=16\text{J}$ , what are the speeds of the particle at the positions listed in part (a)?

(a) Sketch a graph of the potential energy function $U\left(x\right)=k{x}^2\text{/}2+A{e}^{\text{−}\alpha x^2},$ where $k,A,\text{and}\alpha$ are constants. (b) What is the force corresponding to this potential energy? (c) Suppose a particle of mass m moving with this potential energy has a velocity $v_a$ when its position is $x=a$ . Show that the particle does not pass through the origin unless