Problem 30 - Physics for Scientists and Engineers with Modern Physics

Question

(1) A 16.0 -kg child descends a slide 2.20 \(\mathrm{m}\) high and reaches the bottom with a speed of 1.25 \(\mathrm{m} / \mathrm{s} .\) How much thermal energy due to friction was generated in this process?

Step-by-Step Solution

Answer

332.5 J of thermal energy was generated due to friction.

Step 1: Understand the Problem

We are asked to find the thermal energy generated through friction as a child slides down from a height. The initial gravitational potential energy minus the kinetic energy at the bottom of the slide gives us the thermal energy lost.

Step 2: Calculate Initial Potential Energy

The initial potential energy (PE) at the top of the slide is given by the formula: \[ PE = mgh \]where \( m = 16.0 \text{ kg} \), \( g = 9.81 \text{ m/s}^2 \), and \( h = 2.20 \text{ m} \). Substitute the values:\[ PE = 16.0 \times 9.81 \times 2.20 \]

Step 3: Calculate Potential Energy

Perform the calculation for the potential energy:\[ PE = 16.0 \times 9.81 \times 2.20 = 345.0 \text{ J} \]Thus, the initial potential energy is 345.0 J.

Step 4: Calculate Kinetic Energy at the Bottom

The kinetic energy (KE) at the bottom of the slide is given by the formula:\[ KE = \frac{1}{2}mv^2 \] where \( m = 16.0 \text{ kg} \) and \( v = 1.25 \text{ m/s} \). Substitute the values:\[ KE = \frac{1}{2} \times 16.0 \times (1.25)^2 \]

Step 5: Calculate Kinetic Energy

Perform the calculation for the kinetic energy:\[ KE = \frac{1}{2} \times 16.0 \times 1.5625 = 12.5 \text{ J} \]Thus, the kinetic energy at the bottom is 12.5 J.

Step 6: Calculate Thermal Energy due to Friction

The thermal energy due to friction (TE) is calculated by subtracting the kinetic energy from the initial potential energy:\[ TE = PE - KE = 345.0 - 12.5 \]

Step 7: Solve for Thermal Energy

Perform the subtraction to find the thermal energy:\[ TE = 345.0 - 12.5 = 332.5 \text{ J} \]Thus, the thermal energy generated due to friction is 332.5 J.

Key Concepts

Potential Energy

Potential energy is the stored energy of an object due to its position or state. In the context of this problem, when the child is at the top of the slide, they have gravitational potential energy because of their height above the ground. The formula to calculate this is given by

• \( PE = mgh \)

where \( m \) is the mass (16.0 kg in this case), \( g \) is the acceleration due to gravity (9.81 m/s²), and \( h \) is the height (2.20 m). Substituting these values gives a potential energy of 345.0 J. This energy is what the child starts their descent with, ready to be converted as they go down the slide.

Potential energy is a crucial concept in physics because it helps us understand energy storage and conversion in various situations. Here, it helps to eventually calculate how much energy was lost to thermal energy which involves friction.

Kinetic Energy

As the child descends the slide, the potential energy becomes kinetic energy. Kinetic energy is the energy of motion and can be calculated using

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is again the mass of the child, and \( v \) is their velocity (1.25 m/s at the bottom). Plugging in the numbers, the kinetic energy calculates to 12.5 J. This is significantly less than the initial potential energy, telling us not all initial energy converted into motion.

Understanding kinetic energy is essential to analyze how energy is transformed and transferred. In scenarios like this, it helps us figure out what happens to the remaining energy, indicating that other forces, like friction, played a role.

Thermal Energy

Thermal energy, in this context, refers to the heat generated due to friction between the child's clothing and the slide's surface. Thermal energy is not easy to measure directly in such problems, so we find it by subtraction:

• \( TE = PE - KE \)

Here, \( TE \) is the thermal energy, \( PE \) is the initial potential energy, and \( KE \) is the kinetic energy at the bottom. Using 345.0 J for potential energy and 12.5 J for kinetic energy, we calculate that 332.5 J of energy was transformed into thermal energy.

This transformation is critical in physics, as it explains energy loss and how friction impacts systems, converting mechanical energy into heat.

Friction

Friction is a force that opposes motion between two surfaces that are in contact. It plays a significant role in converting some of the mechanical energy into thermal energy, as seen in this problem. Without friction, the child would have gone down the slide without losing speed, converting almost all potential energy into kinetic energy.

Friction works due to surface roughness and induces heat by causing molecular agitation within the materials, hence leading to the concept of 'thermal energy due to friction.' In real-world applications like this slide, friction ensures a safer slide and stops objects from moving infinitely.

• Generates heat
• Slows motion
• Converts mechanical energy

Understanding friction allows us to design better systems that can control the energy flow efficiently.

Physics Problem-solving

Solving physics problems requires a systematic approach to understanding and applying fundamental principles. The solution involves a series of steps to break down complex concepts into manageable parts. This problem, involving a child on a slide, showcases the use of conservation of energy, where

• Potential energy initially stored
• Converts to kinetic energy
• Converts to thermal energy through friction

Using formulas for potential and kinetic energy, we analyzed energy changes as the child descended. Such structured problem-solving is essential not just in physics but any scientific inquiry. It helps develop keen analytical skills useful across scientific disciplines and everyday contexts.