Force & Potential Energy Practice Problems

The binding energy of the single electron in He⁺ is 24.6 eV and the mass of He⁺ is 4.001506 amu. Calculate the percentage decrease in mass when a He²⁺ ion gains an electron to form the He⁺ ion. Express the result in amu and as a percentage of He ²⁺ mass.

Imagine a ball rolling along a horizontal track under the influence of potential energy given by U(x) = Ax² + Bcos((π/2L)x), where A, B, and L are constants. Determine the magnitude and direction of the force acting on the ball when it reaches (i) the starting point ( x = 0), (ii) the midpoint of the track (x = L/2), and (iii) the end of the track (x = L).

Imagine a spaceship cruising along a space-time continuum, traveling on the x-axis. As it moves through space, it experiences a potential energy U = x⁵ - x³ J, where x represents the spaceship's position in meters. Determine if the spaceship is in stable or unstable equilibrium at each of its equilibrium points.

Use the conservation of energy principle to calculate the minimum speed required for the object to travel from point B to point A if the mass of the object is 120g.

Consider a system comprising two particles, X and Y, interacting with each other. Each particle exerts a constant force of 20.0 N on the other. If X moves 2 cm closer to Y while Y moves 4 cm closer to X, what is the change in the potential energy of the system?

Consider a system in which a particle of mass m moves along a track in the range 0 m ≤ x ≤ 2 m, subject to a potential energy function U(x) = (8 J) [1 − cos((2.22 rad/m) x)]. The particle's position is denoted by x. Determine whether each point within this range represents a point of stable or unstable equilibrium.

An object of mass m is confined to move along the x-axis in a one-dimensional system. The particle experiences a  potential energy function given by U(x) = A/x³ - B/x² where A and B are constants (always positive). Determine the equilibrium positions of the particle where the net force on the particle is zero.

Hint: Recall that the net force acting on the particle is the negative gradient of the potential energy function.

An object is moving along the y-axis with a potential energy function U = 5y² + 3, where y is the vertical distance traveled and is in meters. Determine the y-component of the force acting on the object when it is at (i) y = 0 m, (ii) y = 2 m, and (iii) y = 6 m.

A group of scientists has developed a new device called a "twist-o-matic" capable of exerting a force on an object. The force law is given as F_x = -c[(x/2)-x₀)]³, where c is a constant. For simplicity, let's assume that the equilibrium position (x_o) of the twist-o-matic is x_o = 0, then F_x = -c[x/2]³. Determine an expression for the potential energy of a "twist-o-matic" when it is compressed or stretched from its equilibrium position.

Suppose an 800-gram object is initially at point O on a potential energy diagram. The particle is initially at rest, and the diagram illustrates the variation in potential energy as the particle moves from O to P, and Q. Determine the speed of the particle when it passes through points (i) P and (ii) Q.

Suppose a small object weighing only 8 grams is placed at position x = 2m on a potential energy diagram. The object is initially at rest, and the figure below shows how the object's potential energy changes as it moves from its initial position. Calculate the object's maximum velocity and determine the position where that maximum velocity is reached.

The figure shows the potential energy with respect to the position of an object undergoing oscillatory motion between x = 1.0 mm and x = 5.0 mm. With a mass of 4.0 g, what is the maximum speed that this object can achieve?

The graph of the potential energy with respect to the position (along the x-axis) of a particle is shown in the figure below. Given the particle's mass of 800 g and mechanical energy of 16 J, determine the points along the particle's path where it experiences a change in direction, also known as turning points. 

If a particle of mass 450 g is positioned at x = 5.0 m and must never surpass the point at x = 8.0 m, what is the highest speed (v_max) it can achieve along its trajectory?

Hint: Use the Law of Conservation of Energy to determine the maximum speed (v_max)  that a particle should have.

A single particle is moving in an environment that is characterized by a potential energy graph shown in the figure below. Determine the x-component of the force (magnitude and direction) acting on the particle when it is positioned at (i) x = 3 cm,  (ii) 6 cm, and  (iii) 10 cm along its path.

A probe is set to leave Jupiter's atmosphere and needs to reach escape velocity to return data to Earth. With Jupiter's radius given as $6.99\times 10^4$ km and its mass as $1.90\times 10^{27}$ kg, determine the escape velocity from Jupiter's surface.

Determine the escape velocity for a spacecraft in Mars' orbit at its average distance from the Sun ($2.28\times 10^8$ km). Compare this escape velocity with the average orbital speed of Mars around the Sun. Assume the mass of the Sun as $2.0\times 10^{30}$ kg. What factor does the escape velocity differ from Mars' orbital speed?

A bead has a one-dimensional motion. The net force exerted on it is F̄ = -(b/y²) ĵ. Treating b as a constant having SI units and assuming that the bead moves along the y-axis, evaluate the potential energy function U(y) if it is zero at y = 3.0 m or U(3.0m) = 0.

In an oxygen molecule ($O_2$ ), the potential energy between the two oxygen atoms due to both repulsion at short distances and attraction at moderate distances can be modeled by $U(r)=-(a/r^8)+(b/r^{16})$ , where $r$ is the distance between the atoms, and $a$ and $b$ are positive constants reflecting the electron cloud overlap and van der Waals forces, respectively. Identify the value of $r$ at which $U(r)$ is minimized and/or maximized.

Two nitrogen atoms within a diatomic molecule interact through forces that can either attract or repel them, depending on their separation distance. This interaction is captured by the Lennard-Jones potential formula: $F(r)=F_0[2(σ/r)^{13}-(σ/r)^7]$, where $F(r)$ is the force between the atoms, $r$ is their separation distance, and $σ$ and $F_0$ are constants specific to the nitrogen molecule. For nitrogen, $F_0$ is 8.50$\times$10^-11 N, and σ is 3.70$\times$10^-11 m. To understand the energy dynamics of this molecule, derive the expression for the potential energy, $U(r)$.

A conservative force acts on a ball along the y-axis. As a result, it moves along the same axis. Its potential energy versus the y-axis graph is given below. It can be seen that the total energy E > U(y), which implies the speed of the ball is always non-zero. Find the interval(s) of y in which the force on the ball acts toward the positive y direction.

As a result of a conservative force acting parallel to the y-axis on an object, it moves along the same axis. Its energy versus the y-axis graph is given in the figure below. In the graph, it can be seen that the total energy E > U(y), which implies the speed of the object is always non-zero. Find the y values at which the force magnitude is a minimum.

In the figure below the potential energy versus the y-axis graph is plotted for a bead moving along the y-axis and upon which a conservative force acts parallel to the same axis. It can also be seen that the total energy E > U(y), which means the speed of the bead is always non-zero. Find the y value at which the force magnitude is a maximum.

The potential energy between two oxygen atoms in an $O_2$ molecule due to their interaction can be expressed as $U(r)=-(a/r^8)+(b/r^{16})$, with $r$ being the distance between the atoms, and $a$ and $b$ positive constants representing various forces. For what values of $r$ does the force $F$ between the atoms become greater than 0, less than 0, and equal to 0?