Problem 24 - University Physics with Modern Physics

Question

A 2.50-kg block on a horizontal floor is attached to a horizontal spring that is initially compressed 0.0300 m. The spring has force constant 840 N/m. The coefficient of kinetic friction between the floor and the block is \(\mu_k =\) 0.40. The block and spring are released from rest, and the block slides along the floor. What is the speed of the block when it has moved a distance of 0.0200 m from its initial position? (At this point the spring is compressed 0.0100 m.)

Step-by-Step Solution

Answer

The speed of the block is approximately 0.65 m/s.

Step 1: Identify Forces and Energy Types Involved

First, identify all forces acting on the block including the force from the spring and kinetic friction. Also, recognize that the problem involves potential energy stored in the spring and kinetic friction which does mechanical work on the block.

Step 2: Calculate Initial Energy

Calculate the initial potential energy stored in the spring using the equation: \[ U_i = \frac{1}{2} k x_i^2 \]where \(k = 840 \text{ N/m}\) and the initial compression \(x_i = 0.0300 \text{ m}\).

Step 3: Calculate Energy Used by Friction

Determine the work done by kinetic friction as the block moves. Use the formula: \[ W_{friction} = \mu_k \cdot m \cdot g \cdot d \]where \(\mu_k = 0.40\), mass \(m = 2.50 \text{ kg}\), gravitational acceleration \(g = 9.8 \text{ m/s}^2\), and distance \(d = 0.0200 \text{ m}\).

Step 4: Calculate Final Energy with Remaining Spring Compression

Calculate the potential energy remaining in the spring: \[ U_f = \frac{1}{2} k x_f^2 \]where the final compression \(x_f = 0.0100 \text{ m}\).

Step 5: Apply Energy Conservation Principle

Apply the conservation of energy principle assuming initial energy is converted into kinetic energy and overcoming friction:\[ E_{initial} = E_{final} + W_{friction} \]\[ \frac{1}{2} k x_i^2 = \frac{1}{2} m v^2 + W_{friction} + \frac{1}{2} k x_f^2 \]Solve this equation for \(v\), the velocity of the block.

Step 6: Solve for Velocity

Plug in the values into the equation derived from the conservation of energy principle and solve for \(v\), the velocity of the block. Complete the algebra to isolate \(v\) and compute its value.

Key Concepts

Kinetic Friction

When an object moves across a surface, it experiences a force called kinetic friction. This force opposes the motion, slowing the object down. The magnitude of kinetic friction depends on two main factors: the normal force (the force perpendicular to the surface) and the coefficient of kinetic friction, denoted as \(\mu_k\). This coefficient is unique to the materials in contact.

To calculate the force of kinetic friction, use the formula:

• \(F_{friction} = \mu_k \times N\)
• \(N\) is the normal force, which for horizontal surfaces is usually equal to the product of mass \(m\) and gravitational acceleration \(g\), i.e., \(N = m \times g\).

In the exercise, the force of kinetic friction plays a crucial role as it does work against the spring force, affecting the block's velocity as it slides on the floor.

Potential Energy

Potential energy is stored energy due to an object's position or configuration. In this scenario, we're dealing specifically with the "spring potential energy" or elastic potential energy. It's stored in the compressed spring attached to the block. The formula to calculate this is:

• \(U = \frac{1}{2} k x^2\)
• \(k\) is the spring constant, indicating its stiffness, and \(x\) is the compression distance from its equilibrium position.

Initially, the spring is compressed 0.0300 m, storing potential energy that is later converted into kinetic energy and used against friction as the block moves.

Spring Force

Spring force is the force exerted by a compressed or stretched spring upon any object that is attached to it. Hooke's Law describes this linear relationship, which states that the force exerted by the spring is proportional to the distance it is compressed or stretched:

• \(F_{spring} = -k x\)
• The force is always directed opposite to the displacement direction to restore the object to its equilibrium position.

In our problem, this force initially accelerates the block once it is released from its compressed state, translating potential energy into kinetic energy.

Mechanical Work

Mechanical work involves force applied over a distance, transferring energy from one object to another or converting it from one form to another. In the context of this exercise, work is primarily done by friction as it acts over a distance of 0.0200 m to oppose motion.Let's break it down:

• Kinetic friction does work using the formula: \(W_{friction} = F_{friction} \times d\), where \(d\) is the distance over which the force is applied.
• The work done by friction reduces the total mechanical energy available for the block's motion.

Thus, accounting for this work is critical to correctly applying the conservation of energy principle when solving for the block's speed.