Question

How much mechanical energy is lost to friction if a 55.0 -kg skier slides down a ski slope at constant speed of \(14.4 \mathrm{~m} / \mathrm{s}\) ? The slope is \(123.5 \mathrm{~m}\) long and makes an angle of \(14.7^{\circ}\) with respect to the horizontal.

Short answer

Answer: The mechanical energy lost to friction is approximately 34,166 J.

Step-by-step solution

Calculate the gravitational potential energy at the top of the slope

To find the gravitational potential energy at the top of the slope (PE_top), we can use the formula: \(PE_{top} = mgh\) where: - \(m\) is mass of the skier (55.0 kg) - \(g\) is the gravitational acceleration (9.81 m/s^2) - \(h\) is the vertical height of the slope which can be calculated by \(h=l\sin(\theta)\), where \(l\) is the length of the slope (123.5 m) and \(\theta\) is the angle of the slope (14.7 degrees). First, let's find the vertical height \(h\): \(h = 123.5\sin(14.7^{\circ})\) \(h \approx 31.5 \mathrm{~m}\) Now we can calculate the gravitational potential energy at the top of the slope: \(PE_{top} = (55.0 \mathrm{~kg})(9.81 \mathrm{~m/s^2})(31.5 \mathrm{~m})\) \(PE_{top} \approx 17,083 \mathrm{~J}\)

Calculate the gravitational potential energy at the bottom of the slope

Since the skier has reached the bottom of the slope, the vertical height (h) is 0, thus the gravitational potential energy at the bottom of the slope (PE_bottom) is also 0.

Find change in gravitational potential energy

To find the change in gravitational potential energy (\(\Delta PE\)), we can subtract the gravitational potential energy at the bottom of the slope (PE_bottom) from the gravitational potential energy at the top of the slope (PE_top): \(\Delta PE = PE_{top} - PE_{bottom}\) \(\Delta PE = 17,083 \mathrm{~J} - 0 \mathrm{~J}\) \(\Delta PE = 17,083 \mathrm{~J}\)

Calculate the work done by friction

As the skier is sliding down the slope at a constant speed, we can assume the force of gravity pulling the skier down the slope is balanced by the force of friction opposing the skier's motion. We can calculate the work done by friction (W_fric) using the formula: \(W_{fric} = F_{fric} \times d \times \cos{\theta}\) where \(F_{fric}\) is the force of friction, \(d\) is the distance traveled by the skier (the length of slope, 123.5 m), and \(\theta\) is the angle between the force of friction and the distance traveled (180 degrees, as friction is acting opposite to the direction of skier's motion). Since the force of gravity is balanced by the force of friction, we can use the component of gravitational force parallel to the slope as the force of friction: \(F_{fric} = mg\sin\theta\) Now, we can calculate the work done by friction: \(W_{fric} = (55.0 \mathrm{~kg})(9.81 \mathrm{~m/s^2})\sin(14.7^{\circ}) \times 123.5\mathrm{~m} \times \cos(180^{\circ})\) \(W_{fric} \approx -17,083 \mathrm{~J}\)

Calculate the mechanical energy lost to friction

To find the mechanical energy lost to friction (E_lost), we can subtract the change in gravitational potential energy (\(\Delta PE\)) from the work done by friction (W_fric): \(E_{lost} = W_{fric} - \Delta PE\) \(E_{lost} = (-17,083 \mathrm{~J}) - (17,083 \mathrm{~J})\) \(E_{lost} = -34,166 \mathrm{~J}\) The mechanical energy lost to friction is approximately 34,166 J.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is a form of energy an object possesses because of its position in a gravitational field. Every object with mass when raised against the gravitational pull of the Earth gains potential energy. The higher the object, the greater the potential energy. For our skier, this energy can be calculated using the formula:
\[ PE = mgh \]
where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.81 \) meters per second squared on Earth), and \( h \) is the vertical height above the reference point. In the provided exercise, the skier's mass and the height from the top of the slope are used to calculate the initial gravitational potential energy. As the skier descends, the potential energy decreases as it is transformed into other energy forms, principally kinetic and some of it lost as heat due to friction.
To enhance the understanding of this concept, visual aids, such as graphs or diagrams illustrating potential energy at various points along the slope, would be beneficial. Additionally, demonstrations or simulations that show how energy conversions take place could provide students with a more intuitive grasp.

Work-Energy Principle

The work-energy principle is a concept that describes the relationship between the work done on an object and the change in the object's mechanical energy. Work is done when a force is applied to an object and the object moves in the direction of the force. The principle can be simply stated as:
\[ \text{Work done} = \text{Change in energy} \]
In other words, the total work done by all the forces acting on an object results in a change in the object's kinetic energy or potential energy or both. For the skier in our exercise, since the speed is constant, implying no change in kinetic energy, all the work done by friction goes into overcoming the gravitational pull and is hence completely transformed into heat, equating to the energy loss. This is why we calculate the work done by friction over the distance, which is equal in magnitude but opposite in sign to the skier's lost potential energy, yielding the total mechanical energy lost to friction. Demonstrations of kinetic and potential energy changes, using roller coasters or pendulums, could help illustrate the work-energy principle in a more tangible form, making it simpler for students to internalize.

Force of Friction

The force of friction is the force that opposes the relative motion or attempted motion between two surfaces in contact. It is a resistive force that can convert mechanical energy into thermal energy, effectively reducing the total mechanical energy of a moving object. Calculating the force of friction depends on the nature of the surfaces and the normal force exerted between them. For an object sliding down an incline, like our skier, the force of friction is directly related to the component of gravitational force parallel to the slope. Mathematically, this is expressed as:
\[ F_{fric} = mg\sin(\theta) \]
where \( m \) is the mass of the object, \( g \) the acceleration due to gravity, and \( \theta \) the angle of incline. As the skier moves down at a constant speed, it implies that the force of friction is balancing out the downhill component of gravity, indicating that the mechanical energy lost is solely due to this force of friction. Providing real-life examples, such as explaining braking in cars or the slow-down of objects on ramps, can help students relate to the concept of friction in everyday contexts.