Problem 76 - University Physics with Modern Physics

Question

Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. One sphere is 60.0 \(\mathrm{cm}\) in diameter, has mass 50.0 \(\mathrm{g}\) and contains \(-10.0 \mu \mathrm{C}\) of charge. The other sphere is 40.0 \(\mathrm{cm}\) in diameter, has mass 150.0 \(\mathrm{g}\) , and contains \(-30.0 \mu \mathrm{C}\) of charge. Find the maximum acceleration and the maximum speed achieved by each sphere (relative to the fixed point of their initial location in space), assuming that no other forces are acting on them. (Hint: The uniformly distributed charges behave as though they were concentrated at the centers of the two spheres.)

Step-by-Step Solution

Answer

Maximum speed is achieved when the initial potential energy is completely converted into kinetic energy of the spheres.

Step 1: Understanding the Problem

We have two charged spheres with given masses, charges, and diameters. They're initially in contact and release each other due to electrostatic repulsion. We need to find the maximum acceleration and speed each achieves.

Step 2: Calculate Initial Distance of each Center from the Initial Point

Calculate the center of each sphere from the point of contact: since they are in contact, the distance from the point where they touch to each center is half the diameter. Thus, the large sphere's center is 0.3 m (60 cm / 2) away, and the small sphere's center is 0.2 m (40 cm / 2) away.

Step 3: Determine Force Between Spheres

Using Coulomb's law, the force between the spheres is given by:\[ F = \frac{k \, |q_1| \, |q_2|}{d^2} \]where \(k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\), \(q_1 = -10.0 \, \mu \mathrm{C} = -10.0 \times 10^{-6} \, \mathrm{C}\), \(q_2 = -30.0 \, \mu \mathrm{C} = -30.0 \times 10^{-6} \, \mathrm{C}\) and \(d = 0.5 \, \text{m} (0.3 + 0.2)\). Substituting these values, compute the force \(F\).

Step 4: Find Maximum Acceleration Using Newton’s Second Law

Newton’s second law states \( F = ma \). Therefore, the acceleration for each sphere:- For the small sphere, \( a_1 = \frac{F}{m_1} \), where \( m_1 = 0.05 \, \text{kg} \).- For the large sphere, \( a_2 = \frac{F}{m_2} \), where \( m_2 = 0.15 \, \text{kg} \).Compute the acceleration values \( a_1 \) and \( a_2 \).

Step 5: Determine Separation at Maximum Speed

The maximum kinetic energy is equal to the initial potential energy since mechanical energy is conserved. Thus, when the spheres reach their maximum speed, the potential energy due to electrostatic repulsion is completely converted into kinetic energy.

Step 6: Calculate Maximum Speed Using Energy Conservation

Initially, all energy is electrostatic potential energy:\[ U_i = \frac{k \, |q_1| \, |q_2|}{d} \]At maximum speed, kinetic energy is \( T = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 \) and potential energy is zero. Thus, using conservation of energy:\[ U_i = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 \].Solve these equations using the relationship \( m_1v_1 = m_2v_2 \), stemming from conservation of momentum.

Step 7: Solve Equations for Maximum Speed

Substitute \( v_2 = \frac{m_1}{m_2} v_1 \) into the energy conservation equation and solve for \( v_1 \) and then \( v_2 \), using previously calculated values to find the maximum speeds of both spheres.

Key Concepts

Coulomb's Law

Coulomb's Law is essential in electrostatics because it describes the force between two point charges. It tells us that the magnitude of the electrostatic force (\( F \)) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The equation is \[ F = \frac{k \, |q_1| \, |q_2|}{r^2} \]where \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{N m}^2/\text{C}^2) \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the centers of the charges.
In this exercise, we used Coulomb's Law to calculate the force between two spheres with charges of \(-10.0\, \mu \text{C}\) and \(-30.0\, \mu \text{C}\) respectively. This force causes the spheres to repel each other and is crucial in calculating subsequent acceleration and maximum speed.

Newton's Second Law

Newton’s Second Law provides a link between force and motion, stating that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \( F = ma \). This is a powerful tool when calculating how objects respond to forces.
For each of our spheres, once we found the electrostatic force using Coulomb's Law, we implemented Newton’s Second Law to determine acceleration. The smaller sphere with mass \(0.05 \, \text{kg}\) had an acceleration \( a_1 \), and the larger sphere with mass \(0.15 \, \text{kg}\) had an acceleration \( a_2 \). By combining these laws, we tailored our understanding of the system's responses under electrostatic force.

Energy Conservation

Energy Conservation is a cornerstone principle in physics, asserting that energy cannot be created or destroyed but only converted between forms. Here, we analyze how initial electrostatic potential energy transforms into kinetic energy as the spheres move apart.
Initially, the system's energy is all potential, captured by \[ U_i = \frac{k \, |q_1| \, |q_2|}{d} \] with \( d \) being the distance between charges. As the spheres move away, this potential energy converts to kinetic energy, given by \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \). Once the kinetic energy peaks, the potential energy becomes zero, indicating that all potential energy has been converted. Knowing this helped us calculate the maximum speed of the spheres as they sped apart.

Momentum Conservation

Momentum Conservation ensures that within a closed system, the total momentum remains constant, unless acted upon by an external force. In this scenario, the momentum between spheres is shared and constant throughout their motion.
When dealing with the motion of the two spheres, applying momentum conservation gave the equation \( m_1 v_1 = m_2 v_2 \). This relation allowed us to expressed one velocity in terms of the other and solve for the maximum speeds accurately. Utilizing this principle in tandem with Energy Conservation let us engage precisely with how fast each sphere would travel when released.