Problem 4 - An Introduction to Mechanics

Question

A small cube of mass \(m\) slides down a circular path of radius \(R\) cut into a large block of mass \(M\), as shown. \(M\) rests on a table, and both blocks move without friction. The blocks are initially at rest, and \(m\) starts from the top of the path. Find the velocity \(v\) of the cube as it leaves the block.

Step-by-Step Solution

Answer

The velocity is \(v = \sqrt{\frac{2MgR}{M + m}}\).

Step 1: Understand the Physics Concepts Involved

This problem involves conservation of energy and conservation of momentum principles. The small cube finds itself in a potential energy well at the start and will transform this potential energy into kinetic energy as it descends. The large block moves horizontally due to the motion of the small cube, implying the use of conservation of momentum in the horizontal direction.

Step 2: Determine Initial Potential Energy

The initial potential energy when the small block of mass \(m\) is at the top is given by\[ PE_i = mgh = mgR, \]where \(h = R\) is the height from which it starts.

Step 3: Apply Conservation of Energy

When the small block slides down and reaches horizontal position, its potential energy is converted into its kinetic energy and the kinetic energy of the large block. Thus,\[ mgR = \frac{1}{2} mv^2 + \frac{1}{2} MV^2, \]where \(v\) is the velocity of the small cube and \(V\) is the velocity of the large block.

Step 4: Apply Conservation of Momentum

In the horizontal direction, the system's total momentum should be conserved. Initially, there's no momentum horizontally, so\[ mv = MV. \]Equation expresses that the horizontal momentum of the small block equals the horizontal momentum of the large block.

Step 5: Solve for Velocities

Using the momentum conservation equation, express \(V\) in terms of \(v\):\[ V = \frac{mv}{M}. \]Substitute this value of \(V\) into the energy equation:\[ mgR = \frac{1}{2} mv^2 + \frac{1}{2} M\left(\frac{mv}{M}\right)^2. \]Simplify,\[ mgR = \frac{1}{2} mv^2 + \frac{1}{2} \frac{m^2v^2}{M}. \]

Step 6: Solve for the Small Cube's Velocity

Factor out \(v^2\) in the energy equation:\[ mgR = v^2 \left(\frac{m}{2} + \frac{m^2}{2M}\right). \]To solve for \(v^2\), rearrange:\[ v^2 = \frac{2mgR}{m + \frac{m^2}{M}}. \]Simplify further:\[ v^2 = \frac{2MgR}{M + m}. \]Finally, \(v\) is given by:\[ v = \sqrt{\frac{2MgR}{M + m}}. \]

Key Concepts

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics that states that the total momentum of a closed system remains constant if it is not subjected to external forces. In simple terms, momentum is conserved when the forces acting externally are negligible or zero. This principle is particularly useful when analyzing collisions or interactions between two or more objects, as it allows us to predict their motionin a straightforward way.In the context of our exercise involving a small cube and a large block, both objects move without friction, which means there are no external horizontal forces acting on them. Thus, the law of conservation of momentum applies. Initially, the entire system is at rest, so the total horizontal momentum is zero. As the cube descends and moves in the circular path, it gains momentum, and so does the large block in the opposite direction. We can express the conservation of momentum mathematically using the equation: \[ mv = MV \] where:- \( m \) is the mass of the small cube,- \( v \) is the velocity of the small cube, - \( M \) is the mass of the large block,- \( V \) is the velocity of the large block.This indicates that any increase in the cube's momentum must be matched by an equal and opposite change in the block's momentum, maintaining the total momentum at zero, as initially it was.

Potential Energy

Potential energy is the stored energy of position possessed by an object. It represents the potential for doing work. In our exercise, when the small cube is at the top of the circular path, it possesses maximum gravitational potential energy. This energy is due to its height above the reference point, usually taken at the lowest point of the path at the surface of the table.The formula for calculating the initial potential energy \( PE_i \) is given by:\[ PE_i = mgh = mgR \]where:

• \( m \) is the mass of the small cube,
• \( g \) is the acceleration due to gravity, a constant \( 9.8 \, \text{m/s}^2 \),
• \( R \) is the radius of the circular path, which corresponds to the height \( h \) since the cube starts at the top.

As the small cube slides down, this potential energy is converted into kinetic energy. This transformation is at the heart of the conservation of energy principle, which we will delve into in the next sections.Understanding potential energy helps students affirm how energy remains constant within a given system, only changing forms from potential to kinetic and vice versa, as observed in many natural processes.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. As the potential energy of the small cube decreases while it slides down the path, it transforms into kinetic energy. This transition is crucial to describing how fast the small cube moves when it leaves the path.The kinetic energy of an object in motion is calculated using the expression:\[ KE = \frac{1}{2} mv^2 \]In our exercise:

• The symbol \( m \) once again represents the mass of the small cube.
• The symbol \( v \) denotes the velocity it achieves at the bottom.

In this scenario, the large block also receives a share of kinetic energy. When we account for the conservation of energy, we see how the initial potential energy \( mgR \) of the small cube gets split into two components:

• The kinetic energy of the small cube, \( \frac{1}{2} mv^2 \).
• The kinetic energy of the large block, \( \frac{1}{2} MV^2 \), where \( V \) is the velocity at which it moves due to the cube's descent.

Thus, the principle of energy conservation doesn't just predict energy transitions within a single object, but also between interacting systems. This law provides a structural basis for understanding how both the small cube and the large block share and distribute energy as they move without friction.