Question

A horizontal spring with spring constant \(k=17.49 \mathrm{~N} / \mathrm{m}\) is compressed \(23.31 \mathrm{~cm}\) from its equilibrium position. A hockey puck with mass \(m=170.0 \mathrm{~g}\) is placed against the end of the spring. The spring is released, and the puck slides on horizontal ice a distance of \(12.13 \mathrm{~m}\) after it leaves the spring. What is the coefficient of kinetic friction between the puck and the ice?

Short answer

Answer: The coefficient of kinetic friction between the hockey puck and the ice is approximately 0.019.

Step-by-step solution

Calculate the initial potential energy of the spring

The potential energy stored in a compressed spring is given by: \(U_{spring} = \frac{1}{2} kx^2\) where \(k\) is the spring constant and \(x\) is the compression distance. Given \(k=17.49 \mathrm{~N/m}\) and compression distance \(x=0.2331 \, \text{m}\), we can calculate the initial potential energy: \(U_{spring} = \frac{1}{2} (17.49\,\text{N/m})(0.2331\,\text{m})^2 \approx 0.474\,\text{J}\)

Calculate the initial velocity of the puck

As the spring releases its energy to the puck, all of its potential energy will be converted into the puck's kinetic energy (assuming energy is conserved). Therefore, we can find the initial velocity of the puck using the expression of the kinetic energy: \(K = \frac{1}{2} mv^2\) where \(m\) is the mass of the puck and \(v\) is the speed. After converting the mass of the puck to kg (from g to kg), we have \(m=0.17\, \text{kg}\), and now we can solve for \(v\): \(0.474\,\text{J} = \frac{1}{2} (0.17\,\text{kg})v^2\) \(v \approx 4.06\, \text{m/s}\)

Calculate the work done by the friction force

The work done by the friction force acting on the puck is given by \(W_{friction} = -F_{friction} d\) where \(d\) is the distance traveled by the puck after leaving the spring and \(F_{friction} = \mu m g\) (friction force) with \(\mu\) being the coefficient of kinetic friction that we are trying to find, and \(g\) is the acceleration due to gravity (\(9.81\,\text{m/s}^2\)). The negative sign in the work equation indicates that the friction is working against the motion of the puck. Now we can write the equation as: \(W_{friction} = -\mu (0.17\,\text{kg})(9.81\,\text{m/s}^2)(12.13\,\text{m})\)

Equate the work done by friction with the initial kinetic energy to find the coefficient of kinetic friction

As the work done by friction force is equal to the change in kinetic energy, in this case, the initial kinetic energy minus the final kinetic energy (which is zero, because the puck stops): \(W_{friction} = -K \Longrightarrow -\mu (0.17\,\text{kg})(9.81\,\text{m/s}^2)(12.13\,\text{m}) = -0.474\,\text{J}\) Solving for the coefficient of kinetic friction, we get: \(\mu \approx 0.019\) Thus, the coefficient of kinetic friction between the puck and the ice is approximately 0.019.

Key concepts

Potential Energy of a Spring

When dealing with problems related to springs, it's essential to have a good grasp of what potential energy in a spring represents. This form of energy, referred to as elastic potential energy, is stored when a spring is compressed or stretched from its equilibrium position. The amount of energy stored is directly related to the spring's stiffness, characterized by its spring constant (\(k\)), and the displacement from the equilibrium position (\(x\)).

The formula to calculate the potential energy stored in a spring is given by:
\(U_{spring} = \frac{1}{2} kx^2\).

Understanding this concept allows you to solve various problems, such as determining the initial burst of energy that propels an object attached to the spring. In our exercise, knowing that a spring with a constant of 17.49 N/m is compressed by 23.31 cm allows us to calculate this stored energy, which is crucial for the subsequent phases of the problem-solving process. The calculated potential energy is converted into kinetic energy when the spring is released, setting the stage for the hockey puck’s sliding journey across the ice.

Conservation of Mechanical Energy

Conservation of mechanical energy is a fundamental principle in physics, especially when it comes to understanding the behavior of systems involving motion and forces. It states that in an isolated system, with no non-conservative forces like friction or air resistance doing work, the total mechanical energy remains constant.

Therefore, in such systems, the sum of potential energy and kinetic energy at any point is equal to the sum of potential energy and kinetic energy at any other point. In the context of our exercise scenario, when the hockey puck is released from the compressed spring, assuming no energy losses, its initial potential energy (\(U_{spring}\)) is completely transferred into kinetic energy (\(K\)) as the puck starts moving.

\(U_{spring} = K\).

However, when non-conservative forces like friction are in play, they do work on the system which converts mechanical energy into other forms like heat, leading to a reduction in the total mechanical energy. This needs to be considered when applying the conservation principle to real-world problems like the movement of the hockey puck on ice.

Work Done by Friction Force

The work done by a force is a measure of the energy transferred by that force over a distance. When it comes to friction, the work done is usually against the direction of movement, thus removing kinetic energy from the system. This is reflected in the exercise where the hockey puck’s kinetic energy is dissipated by the work done by the frictional force between the puck and the ice.

Mathematically, the work done by friction (\(W_{friction}\)) is the product of the friction force (\(F_{friction}\)), the distance the object moves while the force is applied (\(d\)), and the cosine of the angle between the force and the direction of movement. Since friction opposes the movement, the work by friction is negative:
\(W_{friction} = -F_{friction} d = -\frac{1}{2}mv^2\)

In our example, the friction force depends on the coefficient of kinetic friction, the mass of the puck, and gravity. By equating the work done by friction to the negative of the initial kinetic energy, we can solve for the coefficient of kinetic friction. The negative value indicates energy is taken from the system, slowing the puck until it eventually stops, demonstrating the concept of work done by friction force in a real-life application.