Problem 49 - University Physics with Modern Physics

Question

An experiment is performed in deep space with two uniform spheres, one with mass 50.0 \(\mathrm{kg}\) and the other with mass 100.0 \(\mathrm{kg} .\) They have equal radii, \(r=0.20 \mathrm{m} .\) The spheres are released from rest with their centers 40.0 \(\mathrm{m}\) apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres. (a) Explain why linear momentum is conserved. (b) When their centers are 20.0 m apart, find (i) the speed of each sphere and (ii) the magnitude of the relative velocity with which one sphere is approaching the other. (c) How far from the initial position of the center of the 50.0-kg sphere do the surfaces of the two spheres collide?

Step-by-Step Solution

Answer

Linear momentum is conserved. Speeds: 0.8 m/s (50kg), 0.4 m/s (100kg), Relative velocity: 1.2 m/s. Collision at 13.3 m from initial point of 50 kg sphere.

Step 1: Understanding Linear Momentum Conservation

In an isolated system where no external forces act, the linear momentum is conserved. The gravitational force between the two spheres is internal to the system. Hence, the total linear momentum before and during the motion remains constant. Initially, both spheres are at rest, so the total initial momentum is zero. As they move toward each other, the momentum of one sphere is equal and opposite to that of the other, keeping the total momentum zero.

Step 2: Finding the Speeds Using Energy Conservation

The initial gravitational potential energy of the system is converted into kinetic energy as the spheres move closer. The initial potential energy is:\[ U_i = -\frac{Gm_1m_2}{r_i} \]where \(m_1=50.0\) kg, \(m_2=100.0\) kg, and \(r_i=40.0\) m. The potential energy when their centers are 20.0 m apart:\[ U_f = -\frac{Gm_1m_2}{r_f} \]Kinetic energy is given by:\[ K = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 \]Since momentum \(m_1v_1 = m_2(-v_2)\) (they are moving towards each other), solve for speeds:\[ v_1 = \frac{m_2v_2}{m_1} \]Equating the initial and final energies and substituting in the above, we get equations that can be solved to find \(v_1\) and \(v_2\).

Step 3: Calculating Relative Velocity

The magnitude of the relative velocity of one sphere approaching the other is the sum of their speeds:\[ v_{relative} = v_1 + v_2 \]

Step 4: Finding the Collision Point

At collision, the surfaces of the spheres are touching. The distance between the centers is the sum of their radii, so 0.40 m. Due to conservation of momentum:\[ m_1d_1 = m_2d_2 \]where \(d_1\) and \(d_2\) are the displacements from their initial positions. Find \(d_1\). The distance from the initial position of the 50.0 kg sphere to the collision point is \(r_i - (d_1 + 0.2\text{ m})\).

Key Concepts

Linear Momentum Conservation

Understanding linear momentum conservation is key to solving problems like the one in this experiment. Linear momentum refers to the product of an object's mass and velocity. In an isolated system, where no external forces are acting, the total linear momentum remains constant. This principle is incredibly useful when analyzing the motion of objects under mutual interaction, like the two spheres in our example. Here, the only force at play is the gravitational attraction between the spheres, making it an internal force. Initially, both spheres are at rest, meaning their velocities are zero, and so is the total initial momentum. As they move towards each other, each sphere gains momentum. Yet, because one moves in the opposite direction of the other, their momenta cancel out, keeping the total momentum of the system unchanged at zero. This remarkable balance highlights how even movements manifest without external influence can conserve momentum.

Energy Conservation

Energy conservation is another crucial concept when analyzing the behavior of the spheres. This principle states that the total energy of an isolated system remains constant. It involves converting energy from one form to another without any loss. In this scenario, the initial energy is entirely gravitational potential energy due to the distance between the centers of the spheres. As the spheres move closer, this potential energy starts converting into kinetic energy, which is the energy of motion. The total energy, as per conservation laws, remains constant, allowing us to use an energy equation to find the speeds of the spheres. The key takeaway is that even as the spheres accelerate toward each other, albeit internal forces, the total sum of potential and kinetic energy is preserved. This conservation approach allows the calculation of their speeds when the potential energy changes due to the reduced distance.

Potential Energy

Potential energy, in this context, deals with the gravitational pull between the two spheres. When they are released, the gravitational force has potential energy stored due to their initial separation.The formula to determine gravitational potential energy is:\[ U = -\frac{Gm_1m_2}{r} \]Here, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the spheres, and \( r \) is the distance between their centers.In our problem, when the spheres are initially 40 m apart, they hold a certain potential energy. As they get closer to 20 m apart, the potential energy decreases (becomes more negative) because they are closer, and more energy is now kinetic (motion-related). This shift from potential to kinetic energy as the spheres move toward each other illustrates how energy dynamically transforms while still conserving the system's overall energy status.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. As the spheres draw towards each other, the potential energy converts into kinetic energy. This transformation fuels the speed at which the spheres approach one another.The formula for kinetic energy is:\[ K = \frac{1}{2}mv^2 \]where \( m \) is the mass and \( v \) is the velocity of the object.In the case of the two spheres, calculating their kinetic energy as they meet closer helps determine their speeds. Since both kinetic and potential energies contribute to the total energy, knowing one helps in finding the other. Our conservation equations let us compute how fast each sphere moves by converting the potential energy difference into kinetic energy, illustrating energy's role in changing the states of motion for interacting objects.