Problem 96 - College Physics

Question

\(\bullet\) \(\bullet\) A 68 kg skier approaches the foot of a hill with a speed of 15 \(\mathrm{m} / \mathrm{s} .\) The surface of this hill slopes up at \(40.0^{\circ}\) above the horizontal and has coefficients of static and kinetic friction of 0.75 and \(0.25,\) respectively, with the skis. (a) Use energy con- servation to find the maximum height above the foot of the hill that the skier will reach. (b) Will the skier remain at rest once she stops, or will she begin to slide down the hill? Prove your answer.

Step-by-Step Solution

Answer

The maximum height is approximately 7.03 m; the skier will slide down.

Step 1: Identify Initial and Final Energies

The initial energy of the skier consists of kinetic energy since the skier is at the base of the hill. The final energy is potential energy when the skier reaches maximum height. The frictional force will do work against the skier's motion. We use the conservation of mechanical energy with work done by friction taken into account.

Step 2: Calculate Initial Kinetic Energy

Initial kinetic energy is given by the formula: \[ KE_i = \frac{1}{2} m v^2 \]Substitute the given values: \[ KE_i = \frac{1}{2} \times 68 \times (15)^2 = 7650 \, \text{Joules} \]

Step 3: Calculate Work Done by Friction

The work done by friction is calculated as:\[ W_f = f_k \times d \]Where \(f_k\) is the kinetic friction force, given by \(f_k = \mu_k \times N\) and \(N\) is the normal force. On an incline, \(N = mg \cos \theta\). Thus, \[ f_k = 0.25 \times 68 \times 9.8 \times \cos(40^\circ) \approx 128.55 \, \text{N} \]Since distance \(d\) along the slope is unknown, we later know from energy equations that:\[ W_f = f_k \times h/\sin(40^\circ) \]

Step 4: Apply Conservation of Energy

According to the conservation of energy principle:\[ KE_i = PE_f + W_f \]Where \(PE_f = mgh\) is the potential energy at the maximum height, and \(W_f\) is the work done by friction:\[ 7650 = 68 \times 9.8 \times h + 128.55 \times h/\sin(40^\circ) \]Solve for \(h\):\[ 7650 = 68 \times 9.8 \times h + (128.55/\sin(40^\circ)) \times h \]Thus,\[ 68 \times 9.8 + 128.55/\sin(40^\circ) = 1089 \approx 1089h \, (after \ solving) \, \Rightarrow h \approx \frac{7650}{1089} \approx 7.03 \, \text{m} \]

Step 5: Determine If Skier Remains at Rest

To check if the skier remains at rest, compare the static friction force with the gravitational component along the incline. The gravitational component along the incline is given by:\[ f_g = mg \sin \theta \approx 68 \times 9.8 \times \sin(40^\circ) \approx 430.9 \, \text{N} \]The maximum static friction force is:\[ f_{s, max} = \mu_s \times N = 0.75 \times 68 \times 9.8 \times \cos(40^\circ) \approx 385.6 \, \text{N} \]Since \( f_g > f_{s,max} \), the skier will start sliding back.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is a fundamental concept in physics and plays a crucial role in understanding how objects move. The formula for kinetic energy is given by \[ KE = \frac{1}{2} m v^2 \]where:

• \( m \) is the mass of the object in kilograms (kg),
• \( v \) is the velocity of the object in meters per second (m/s).

In the context of the skier exercise, when the skier is at the base of the hill, she is moving with a speed of 15 m/s, so her kinetic energy is at its maximum. As she moves up the hill, her speed decreases due to gravity and friction, converting her kinetic energy into potential energy. Understanding kinetic energy helps to predict how the skier will behave as she moves up the incline.

Potential Energy

Potential energy is the energy stored in an object due to its position relative to some reference point, typically the Earth’s surface. In most physics problems, potential energy is associated with gravitational force. The formula for gravitational potential energy is\[ PE = mgh \]where:

• \( m \) is the mass of the object (in kg),
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth),
• \( h \) is the height above the reference point (in meters).

As the skier climbs the hill, her potential energy increases, since she is gaining height. This comes at the expense of her kinetic energy, following the principle of the conservation of energy, which states energy can't be created or destroyed, only transformed. When the skier stops climbing, all her initial kinetic energy will have turned into potential energy and work done by friction.

Frictional Force

Frictional force, often just called friction, is the force resisting the relative motion of solid surfaces and materials sliding against each other. It’s a force that opposes motion between two surfaces that are in contact. Friction is split into two main types: static and kinetic friction.For the skier, the force of friction acts against her as she moves up the hill. The frictional force does work on the system, converting some of the skier's mechanical energy into thermal energy, and plays a critical role in determining how far up the hill she will travel before stopping. Calculating the work done by friction involves finding the kinetic friction force using constants given, such as the coefficient of kinetic friction \( \mu_k \) and the normal force \( N \).

Static and Kinetic Friction

Static friction is the force that keeps an object at rest when a force is applied, up to the point of movement. It is generally stronger than kinetic friction, which takes over once the object is in motion. Here’s how they’re defined:

• Static Friction: The force that must be overcome to start moving an object at rest. Calculated as \( f_{s,max} = \mu_s \times N \), where\( \mu_s \) is the coefficient of static friction.
• Kinetic Friction: The force opposing the motion of an object that is already moving. Calculated as \( f_k = \mu_k \times N \), where \( \mu_k \) is the coefficient of kinetic friction.

In the skier scenario, the static friction determines whether she will remain stationary once she reaches a complete stop. If the downhill component of gravity \( mg \sin \theta \) is greater than \( f_{s,max} \), she will start sliding again. Since the calculated gravitational force component is greater than the maximum static friction, the skier will indeed begin to slide back down.