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 cm in diameter, has mass 50.0 g, and contains \(-\)10.0 \(\mu\)C of charge. The other sphere is 40.0 cm in diameter, has mass 150.0 g, and contains \(-\)30.0 \(\mu\)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.)

Short answer

The maximum accelerations are determined using Coulomb's law and Newton's second law, while velocity is found through the conservation of momentum.

Step-by-step solution

Identify Initial Conditions

We have two spheres with charges \( Q_1 = -10.0 \ \mu C \) and \( Q_2 = -30.0 \ \mu C \). The diameters are 60.0 cm and 40.0 cm respectively, which means their radii are 30.0 cm and 20.0 cm. Their masses are 50.0 g and 150.0 g.

Understand Coulomb's Law

From Coulomb's law, the force \( F \) between two charges is given by \[ F = \frac{k \cdot |Q_1 \cdot Q_2|}{r^2}, \]where \( k = 8.99 \times 10^9 \ N m^2/C^2 \) is Coulomb's constant, and \( r \) is the distance between their centers when they are in contact, \( r = (30.0 + 20.0)/100 = 0.50 \ m \).

Calculate the Force of Repulsion

Substitute the values into Coulomb's law to find the force:\[ F = \frac{8.99 \times 10^9 \ N m^2/C^2 \cdot 10.0 \times 10^{-6} C \cdot 30.0 \times 10^{-6} C}{(0.50 \ m)^2}. \]Solve for \( F \) to find the force of repulsion between the spheres.

Determine Maximum Acceleration

The maximum acceleration \( a \) of each sphere can be found using Newton's second law \( F = ma \). For sphere 1:\[ a_1 = \frac{F}{m_1} \] and for sphere 2:\[ a_2 = \frac{F}{m_2} \].Using previously calculated force \( F \), substitute the respective masses.

Use Conservation of Momentum to Find Relationship Between Speed

Initially, both spheres are at rest, so the total initial momentum is zero. For two spheres moving apart, \[ m_1 \cdot v_1 = m_2 \cdot v_2 \].This equation can be used to relate their maximum speeds, assuming no external forces act on the system.

Calculate Maximum Speeds

Use the relations and conservation laws to solve simultaneously for the maximum speeds \( v_1 \) and \( v_2 \) for each sphere, seeing that initially their starting speed was zero and they both repelled outwards with respect to each other until reaching maximum speeds.

Key concepts

Charge Distribution

In the realm of electrostatics, understanding charge distribution is crucial. Charges within objects, like our plastic spheres, can be distributed in various ways - uniformly or non-uniformly. For our problem, charges are distributed uniformly throughout the interior of each sphere. This means charge is evenly spread across their volume, making calculations simpler.

The hint in the exercise highlights that uniformly distributed charges behave as if they are concentrated at their centers. This is an essential feature in simplification: instead of complex integrals calculating force over volumes, we calculate it as though all charge is at a point - the center of the sphere. This assumption, backed by symmetrical charge distribution, streamlines applying Coulomb's Law to spherical objects. Given the symmetrical and uniform nature, computation involves using just the distance between centers and known charges on each sphere.

• Uniform distribution means equal charge spread, simplifying calculations.
• We treat entire charge as a point at the sphere's center.
• This aids in concise application of electrostatic equations.

Newton's Second Law

Newton's Second Law teaches us about the relationship between force and motion. It states that the force on an object is directly proportional to the acceleration of that object, given by the equation: \( F = ma \).
In our scenario, the force is the electrostatic repulsion calculated using Coulomb's Law. Each sphere accelerates away from their point of contact due to this force. The acceleration experience by each sphere depends on its mass, which varies greatly given the 50.0 g and 150.0 g masses of the spheres.

When calculating the maximum acceleration for each sphere, we substitute the repulsive force for \( F \) and each sphere’s mass for \( m \). The result tells us how quickly each sphere will begin to move away from each other initially. Because they have different masses, their accelerations will not be identical — the lighter sphere will experience greater acceleration.

• Force causes acceleration based on \( F = ma \).
• Electrostatic forces propel each sphere.
• Lighter spheres accelerate more under identical force.

Conservation of Momentum

The principle of conservation of momentum is pivotal when dealing with isolated systems. It posits that the total momentum of a closed system remains constant when no external forces are acting. Here, both spheres start from rest, so their initial total momentum is zero.

As they repel each other, they move in opposite directions. Since no external forces influence them, their combined momentum stays at zero. Mathematically, this is expressed as \( m_1 \cdot v_1 = m_2 \cdot v_2 \). Using this, we can relate the velocities of the spheres, despite not knowing them initially.

Importantly, the conservation of momentum helps find the relationship between their speeds after the force has acted. By rearranging and solving the momentum equation, you can calculate the maximum speed for each sphere.

• Momentum conservation keeps total initial and final momentum equal.
• Opposing velocities counterbalance to maintain zero total momentum.
• Watch the mass-speed interplay to derive speed relations.

Electrostatics Problem Solving

Problem solving in electrostatics involves applying fundamental principles to understand how charged objects interact. Given the forces calculated with Coulomb’s Law, further steps rely on dynamics principles like those of Newton and momentum conservation.

To tackle such problems:

• Start by identifying charge values and their distribution.
• Apply Coulomb’s Law to find the force between charges.
• Use Newton's Second Law to connect forces to acceleration.
• Incorporate conservation laws to tie motion characteristics together.

Each formula has significance. Coulomb's formula provides force magnitudes, Newton's law translates force into motion, and momentum principles harness those motions into quantifiable velocities. Practicing such steps ensures a robust understanding of electrostatic problems and prepares you for various scenarios involving static charges and forces.