Problem 2 - An Introduction to Mechanics

Question

A block of mass \(M\) slides along a horizontal table with speed \(v_{0}\). At \(x=0\) it hits a spring with spring constant \(k\) and begins to experience a friction force, as indicated in the right-hand sketch. The coefficient of friction is variable and is given by \(\mu=b x\), where \(b\) is a constant. Find the distance \(l\) the block travels before coming to rest.

Step-by-Step Solution

Answer

The distance \(l\) the block travels is \( l = \sqrt{\frac{Mv_0^2}{k + bMg}} \)."

Step 1: Identify the forces involved

The block experiences a force due to the spring and a frictional force. The spring exerts a force of magnitude \[ F_{spring} = -kx \]where \( x \) is the compression of the spring. The frictional force is given by:\[ F_{friction} = \mu Mg = bxMg \]

Step 2: Set up the energy equation

The initial kinetic energy of the block is \[ KE_{initial} = \frac{1}{2}Mv_0^2 \]. As it moves, work is done against the frictional force and the spring stores potential energy. At rest, the kinetic energy is zero. Thus, the energy conservation equation is:\[ \frac{1}{2}Mv_0^2 = \frac{1}{2}kx^2 + \int_{0}^{l} bx M g \, dx \]

Step 3: Calculate work done by friction

The work done by the frictional force as the block moves from \(x = 0\) to \(x = l\) is:\[ W_{friction} = \int_{0}^{l} bx M g \, dx = \frac{bMgl^2}{2} \]

Step 4: Substitute and simplify the energy equation

Substitute the expression for friction work into the energy balance equation:\[ \frac{1}{2}Mv_0^2 = \frac{1}{2}kl^2 + \frac{bMgl^2}{2} \].Simplifying and factoring out common terms gives:\[ \frac{1}{2}Mv_0^2 = \frac{l^2}{2}(k + bMg) \]

Step 5: Solve for the distance \(l\)

Isolate \(l\) in the equation:\[ l^2 = \frac{Mv_0^2}{k + bMg} \].Thus:\[ l = \sqrt{\frac{Mv_0^2}{k + bMg}} \].

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It depends on two factors: the mass of the object and its velocity. When an object moves, it gains kinetic energy, which can be calculated using the formula:

• \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass, and \( v \) is the velocity of the object.

In our exercise, the block starts with an initial velocity \( v_0 \) and a mass \( M \), resulting in an initial kinetic energy of \( KE_{initial} = \frac{1}{2} M v_0^2 \).
As the block moves along the table and hits the spring, some of this kinetic energy gets converted into potential energy and work done against friction.
When the block comes to rest, all of its kinetic energy has been transformed into other forms of energy, and the final kinetic energy is zero.

Potential Energy

Potential energy is the energy stored within an object due to its position or configuration. In the context of mechanics, particularly with springs, it is often termed elastic potential energy, which depends on the compression or extension of the spring.

• The potential energy stored in a compressed or stretched spring is given by:\( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the spring's equilibrium position.

In our problem, as the block compresses the spring, the energy from its motion is stored as potential energy in the spring.
The block loses kinetic energy as it goes into compressing the spring.
This potential energy formulated as \( \frac{1}{2} k x^2 \) describes how the spring holds energy that can exert a force back on the block.

Friction Force

Friction force is a resisting force that surfaces exert when they come into contact with each other. It opposes the motion or attempted motion of an object across a surface.

• Friction can sometimes vary, like in our case, where it depends on the distance \( x \) that the block has traveled.
• We use the given expression for the friction coefficient \( \mu = b x \) to find the friction force as \( F_{friction} = bxMg \).

The total work done by this friction force as the block moves to rest contributes to the transition from kinetic to stored potential energy.
When the block progresses along the surface, the friction work is computed through the integral \( \int_{0}^{l} bx M g \, dx \), which when evaluated produces the work done \( W_{friction} = \frac{bMgl^2}{2} \).
This is the energy lost due to the friction that helps bring the block to rest over the distance \( l \).