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Chapter 6: Problem 29

A 75.0 -kg skier rides a 2830 -m-long lift to the top of a mountain. The lift makes an angle of \(14.6^{\circ}\) with the horizontal. What is the change in the skier's gravitational potential energy?

Short Answer

Expert verified
The change in the skier's gravitational potential energy is approximately 525935 J.

Step by step solution

01

Understanding Potential Energy

Gravitational potential energy is given by the formula \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height above the reference point.
02

Determine the Height Gained

To find the height \( h \) gained by the skier, use the length of the lift and the sine of the angle of inclination: \[ h = L \cdot \sin(\theta) \] where \( L = 2830 \, \text{m} \) is the length of the lift and \( \theta = 14.6^{\circ} \) is the angle with the horizontal. Thus, \( h = 2830 \cdot \sin(14.6^{\circ}) \).
03

Calculate the Height

Compute the height \( h \) using the sine function: \( h = 2830 \times \sin(14.6^{\circ}) \approx 2830 \times 0.252 \). So, \( h \approx 713.56 \ \, \text{m} \).
04

Calculate Potential Energy Change

Substitute the known values into the potential energy formula: \[ \Delta PE = mgh = 75.0 \cdot 9.81 \cdot 713.56 \]. Calculate this to determine the change in gravitational potential energy.
05

Final Calculation of Potential Energy

Perform the multiplication: \( \Delta PE = 75.0 \times 9.81 \times 713.56 \approx 525934.77 \, \text{J} \). Therefore, the change in the skier's gravitational potential energy is approximately \( 525935 \, \text{J} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy Formula
Gravitational potential energy is an important concept in physics. It represents the energy an object possesses due to its position in a gravitational field. The potential energy formula is given by:
  • \( PE = mgh \)
In this formula, \( m \) stands for mass, \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \), and \( h \) is the height above the reference point. The greater the height or mass, the more potential energy an object will have.

The unit of potential energy is joules (J). Understanding this concept helps in analyzing how energy is transformed, like when a skier rises to a higher point on a mountain lift, increasing their potential energy.
Height Calculation
Determining the height gained is crucial for calculating gravitational potential energy. In scenarios involving inclined planes, such as a ski lift, the height can be calculated using trigonometry. The basic approach is:

To find height \( h \) gained:
  • Use the formula: \( h = L \cdot \sin(\theta) \)
Where \( L \) is the length of the lift and \( \theta \) is the angle of inclination. Plug in the specific values relevant to the problem to get:
  • \( h = 2830 \times \sin(14.6^{\circ}) \)
  • This simplifies to \( h \approx 713.56 \, \text{m} \)
Calculating the precise height reached helps in determining the exact change in potential energy.
Inclined Plane
Inclined planes are surfaces sloped at an angle, helping objects rise to a height more gradually. They are commonly encountered in physics challenges, such as skiing scenarios. Inclined planes reduce the force needed to lift an object vertically, providing a practical way to achieve greater heights.

When dealing with inclined planes:
  • The incline angle \( \theta \) affects both the climb and energy calculations.
  • The length of the incline \( L \) is particularly essential in height calculations.
  • These concepts are key to understanding how potential energy builds as objects move along inclined surfaces.
Learning about inclined planes illustrates how different forces work together, providing a clearer picture of physics in motion.
Trigonometry in Physics
Trigonometry is a mathematical tool that aids in solving physical problems involving angles and distances. It is particularly useful on inclined planes and in scenarios similar to the skier's lift scenario. Key functions like sine, cosine, and tangent help in resolving components of vectors and calculating heights.

For this problem:
  • The sine function is used: \( \sin(\theta) \), where \( \theta \) is the angle of the incline.
  • Sine relates the angle to the ratio of the opposite side (height gained) over the hypotenuse (length of the lift).
Understanding trigonometry in physics offers insights into complex problems, enabling the calculation of unknown lengths and angles efficiently. This knowledge is essential in many applications, such as engineering, architecture, and various physical sciences.

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Most popular questions from this chapter

A fighter jet is launched from an aircraft carrier with the aid of its own engines and a steam-powered catapult. The thrust of its engines is \(2.3 \times 10^{5} \mathrm{~N}\). In being launched from rest it moves through a distance of \(87 \mathrm{~m}\) and has a kinetic energy of \(4.5 \times 10^{7} \mathrm{~J}\) at liftoff. What is the work done on the jet by the catapult?

A pitcher throws a \(0.140\) -kg baseball, and it approaches the bat at a speed of \(40.0 \mathrm{~m} / \mathrm{s}\). The bat does \(W_{n c}=70.0 \mathrm{~J}\) of work on the ball in hitting it. Ignoring air resistance, determine the speed of the ball after the ball leaves the bat and is \(25.0 \mathrm{~m}\) above the point of impact.

A projectile of mass \(0.750 \mathrm{~kg}\) is shot straight up with an initial speed of \(18.0 \mathrm{~m} / \mathrm{s}\). (a) How high would it go if there were no air friction? (b) If the projectile rises to a maximum height of only \(11.8 \mathrm{~m}\), determine the magnitude of the average force due to air resistance.

Interactive Solution 6.33 at presents a model for solving this problem. A slingshot fires a pebble from the top of a building at a speed of \(14.0 \mathrm{~m} / \mathrm{s}\). The building is \(31.0 \mathrm{~m}\) tall. Ignoring air resistance, find the speed with which the pebble strikes the ground when the pebble is fired (a) horizontally, (b) vertically straight up, and (c) vertically straight down.

A swing is made from a rope that will tolerate a maximum tension of \(8.00 \times 10^{2} \mathrm{~N}\) without breaking. Initially, the swing hangs vertically. The swing is then pulled back at an angle of \(60.0^{\circ}\) with respect to the vertical and released from rest. What is the mass of the heaviest person who can ride the swing?
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