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Chapter 5: Problem 65

A roller-coaster car of mass \(1.50 \times 10^{4} \mathrm{~kg}\) is initially at the top of a rise at point \((1)\). It then moves \(35.0 \mathrm{~m}\) at an angle of \(50.0^{\circ}\) below the horizontal to a lower point (B). (a) Find both the potential energy of the system when the car is at points ( 8 and \((\) B) and the change in potential energy as the car moves from point (A) to point (B), assuming \(y=0\) at point (B). (b) Repeat part (a), this time choosing \(y=0\) at point \(\mathbb{C}\), which is another \(15.0 \mathrm{~m}\) down the same slope from point (B),

Short Answer

Expert verified
For part (a), the potential energy at point A is a certain value calculated in Joules, at point B it is 0 Joules, and the change in potential energy is simply the negative value of the potential energy at point A also given in Joules. For part (b), the potential energy at point A and B increases and is given in Joules, but the change in potential energy remains the same as part (a).

Step by step solution

01

Calculate the potential energy at point A (PE_A)

To find the potential energy at point A, use the formula for gravitational potential energy \( PE = mgh \). The mass \( m \) is given as \( 1.50 \times 10^{4} \) kg, and since it's on top of the rise, the height \( h \) can be calculated using the given distance (35.0 m) and angle (50.0 degrees) and the sine trigonometric function \( h = d \cdot \sin(\theta) \). Putting these into the equation gives \( PE_A = (1.50 \times 10^{4} \mathrm{ kg}) \cdot (9.8 \mathrm{ m/s}^{2}) \cdot (35.0\mathrm{m} \cdot \sin(50.0^{\circ})) \)
02

Find potential energy at point B (PE_B)

Since we define the potential energy at point B to be zero (as stated in the exercise), \( PE_B = 0 \mathrm{ J} \)
03

Calculate the change in potential energy

The change in potential energy is the difference between the potential energy at point B and that at point A, which is \( \Delta PE = PE_B - PE_A = 0 - PE_A \). The negative sign indicates a loss in potential energy as the car moves from A to B.
04

Calculate potential energy for part (b)

For part (b), repeat steps 1, 2 and 3, but this time setting the zero potential energy level at a point C which is another \15.0 m\ down the slope from point B. This increases the height at point A by an additional 15 m, and at point B by 15 m, causing an increase in both potential energy values. The change in potential energy remains the same as before.

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Key Concepts

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

Gravitational Potential Energy
Gravitational potential energy is a form of energy associated with the position of an object in a gravitational field. It depends on the object's mass, the height it is at, and the gravitational acceleration (usually approximated as 9.8 m/s虏 on Earth). The formula used to calculate gravitational potential energy is: \( PE = mgh \), where:
  • \( m \) is the mass of the object in kilograms.
  • \( g \) is the acceleration due to gravity.
  • \( h \) is the height above a chosen reference point.
In the context of our roller-coaster exercise, the potential energy changes as the coaster moves from point A to point B due to the difference in height. The change is calculated using trigonometry to find the vertical component of the car's descent.
Roller Coaster Physics
Roller coaster physics primarily relies on the conversion and conservation of energy. A coaster is raised to a certain height, gaining gravitational potential energy, which converts into kinetic energy (energy of motion) as it descends. In our scenario, as the car moves from point A to point B, it converts potential energy into kinetic energy, causing it to accelerate.

The roller-coaster ride is a practical example of how potential and kinetic energy interchange during motion. The changes in height and slopes experienced by the coaster are the key factors determining its velocity and acceleration at various points.
Energy Conservation
The law of conservation of energy states that energy can neither be created nor destroyed; it can only be transferred or converted from one form to another. In a closed system (like our roller coaster scenario, neglecting air resistance and friction), the total energy remains constant. This means the sum of potential and kinetic energy at any point in the coaster's path should be the same as at any other point.

During its descent from A to B, the roller coaster loses potential energy while gaining an equal amount of kinetic energy. If all energy types are accounted for, there will be no net change in the total energy of the system.
Trigonometry in Physics
Trigonometry plays a crucial role in physics, particularly in finding components of forces or distances along specific angles. In our roller coaster problem, trigonometry is used to determine the vertical height change as the coaster moves along a sloped path.

By using the sine function, which relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle, we can calculate the height change. This is computed as \( h = d \cdot \sin(\theta) \), where \( d \) is the distance traveled by the coaster and \( \theta \) is the angle of descent. This calculation is essential for determining the gravitational potential energy at different points along the track.

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Most popular questions from this chapter

A raw egg can be dropped from a third-floor window and land on a foam-rubber pad on the ground without breaking. If a \(75.0\) -g egg is dropped from a window located \(32.0 \mathrm{~m}\) above the ground and a foam-rubber pad that is \(15.0 \mathrm{~cm}\) thick stops the egg in \(9.20 \mathrm{~ms}\), (a) by how much is the pad compressed? (b) What is the average force exerted on the egg after it strikes the pad? Note: Assume constant upward acceleration as the egg compresses the foam-rubber pad.

Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is 1 kilocalorie, which we define in Chapter 11 as \(1 \mathrm{kcal}=4186 \mathrm{~J}\). Metabolizing 1 gram of fat can release \(9.00 \mathrm{kcal}\), A student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football stadium as fast as she can and as many times as necessary. Is this in itself a practical way to lose weight? To evaluate the program, suppose she runs up a flight of 80 steps, each \(0.150 \mathrm{~m}\) high, in \(65.0 \mathrm{~s}\). For simplicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human muscles is \(20.0 \%\). This means that when your body converts \(100 \mathrm{~J}\) from metabolizing fat, \(20 \mathrm{~J}\) goes into doing mechanical work (here, climbing stairs). The remainder goes into internal energy. Assume the student's mass is \(50.0 \mathrm{~kg}\). (a) How many times must she run Lhe flight of stairs to lose 1 pound of fat? (b) What is her average power output, in watts and in horsepower, as she is running up the stairs?

Starting from rest, a \(5.00-\mathrm{kg}\) block slides \(2.50 \mathrm{~m}\) down a rough \(30.0^{\circ}\) incline. The coefficient of kinetic friction between the block and the incline is \(\mu_{k}=0.496 .\) Determine (a) the work done by the force of gravity, (b) the work done by the friction force between block and incline, and (c) the work done by the normal force. (d) Qualitatively, how would the answers change if a shorter ramp at a steeper angle were used to span the same vertical height?

An \(80.0-\mathrm{kg}\) skydiver jumps out of a balloon at an altitude of \(1000 \mathrm{~m}\) and opens the parachute at an altitude of \(200.0 \mathrm{~m} .\) (a) Assuming that the total retarding force on the diver is constant at \(50.0 \mathrm{~N}\) with the parachute closed and constant at \(3600 \mathrm{~N}\) with the parachute open, what is the speed of the diver when he lands on the ground? (b) Do you think the skydiver will get hurt? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is \(5.00 \mathrm{~m} / \mathrm{s}\) ? (d) How realistic is the assumption that the total retarding force is constant? Explain.

A weight lifter lifts a \(350-\mathrm{N}\) set of weights from ground level to a position over his head, a vertical distance of \(2.00 \mathrm{~m}\). How much work does the weight lifter do, assuming he moves the weights at constant speed?
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