Problem 12 - College Physics

Question

A hockey puck with a mass of \(200 \mathrm{~g}\) and an initial speed of \(25.0 \mathrm{~m} / \mathrm{s}\) slides freely to rest in the space of \(100 \mathrm{~m}\) on a sheet of horizontal ice. How many forces do nonzero work on it as it slows: (a) (1) none, (2) one, (3) two, or (4) three? Explain. (b) Determine the work done by all the individual forces on the puck as it slows.

Step-by-Step Solution

Answer

(a) One force does nonzero work; it's friction. (b) The work done is 62.5 J.

Step 1: Understanding the Motion and Forces

To solve this problem, first recognize that the puck is initially in motion due to its initial speed and then comes to a stop. The forces at work are responsible for the puck's deceleration. Its motion stops due to friction between the puck and the ice, which is a horizontal surface. Friction is the force that is doing work on the hockey puck, slowing it down to a stop. There are no other forces doing significant work. Therefore, the answer to part (a) is option (2): one force is doing nonzero work, which is friction.

Step 2: Identifying Work Done by Friction

Next, identify the nature of work. The work done by the force is given by the equation: \( W = F imes d \)where \( W \) is the work done, \( F \) is the force, and \( d \) is the distance over which the force is applied. Here, the force is the frictional force, and \( d = 100 \text{ m} \).

Step 3: Using Kinematic Equation to Find Frictional Force

Since the initial kinetic energy is converted into work done by friction, use the work-energy principle: the work done by friction is equal to the change in kinetic energy.The initial kinetic energy \( KE_i = \frac{1}{2} m v^2 \) and the final kinetic energy \( KE_f = 0 \).The change in kinetic energy is: \( \Delta KE = KE_f - KE_i = 0 - \frac{1}{2} m v^2 \). Substituting the given values:\( KE_i = \frac{1}{2} \times 0.2 \text{ kg} \times (25.0 \text{ m/s})^2 \).Calculate this to find \( KE_i \).

Step 4: Calculate the Initial Kinetic Energy

Calculate the initial kinetic energy:\( KE_i = \frac{1}{2} \times 0.2 \times 625 = 62.5 \text{ J} \).Thus, the work done by friction is \( 62.5 \text{ J} \), because all initial kinetic energy is converted to work done against friction. So, the work done by the force (friction) is \( 62.5 \text{ J} \).

Key Concepts

Friction

Friction is a force that opposes the motion of objects sliding against each other. In this exercise, the frictional force is what causes the hockey puck to come to a complete stop. When two surfaces, like the puck and the ice, come into contact, microscopic irregularities on the surfaces interact, resulting in friction. This interaction converts the puck's kinetic energy into heat.

• Kinetic Friction: This type of friction occurs when two objects are in relative motion. Here, it acts on the hockey puck sliding over ice, generating a force that opposes its motion.
• Role of Friction: By acting opposite to the direction of motion, friction does work on the puck, slowing it down until all its kinetic energy is dissipated.

Understanding friction is essential for explaining how and why objects in motion eventually come to rest unless other forces are applied.

Kinetic Energy

Kinetic energy (\( KE \) ) is the energy that an object possesses due to its motion. The kinetic energy of an object 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 this scenario, the hockey puck initially has kinetic energy because it is moving at a velocity of 25 m/s.

• Initial Kinetic Energy: For the puck with a mass of 0.2 kg and initial speed of 25 m/s, the kinetic energy is calculated as \( 62.5 \text{ J} \).
• Final Kinetic Energy: When the puck comes to rest, its velocity becomes zero, and thus its kinetic energy is \( 0 \text{ J} \).

The transition from initial to final kinetic energy helps us understand the work done by friction on the puck, as the initial kinetic energy is converted entirely into work overcoming the frictional force.

Work-Energy Principle

The work-energy principle provides a foundation for understanding how forces like friction do work on an object and change its energy state. This principle states that the work done by all forces acting on an object is equal to the change in its kinetic energy. In simpler terms: \[W = \Delta KE\] where \( W \) is the work done and \( \Delta KE \) is the change in kinetic energy.

• Application: In the hockey puck scenario, the frictional force does work on the puck as it slides across the ice.
• Energy Transformation: The entire initial kinetic energy (62.5 J) of the puck is transformed into work done against the frictional force, which ultimately brings the puck to rest.

Using the work-energy principle bridges our understanding of how energy is conserved and transformed from one form (kinetic energy) into the work done against forces like friction, explaining the natural deceleration and stopping of objects in motion.