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Chapter 10: Problem 22

A 72 kg bike racer climbs a 1200 -m-long section of road that has a slope of \(4.3^{\circ} .\) By how much does his gravitational potential energy change during this climb?

Short Answer

Expert verified
The gravitational potential energy change during this climb is 63504 J.

Step by step solution

01

Convert slope to radians

First, convert the given slope from degrees to radians because trigonometric functions in most calculators use radians. To convert degrees to radians, use the formula \(\theta_{radians} = \theta_{degrees} * (\pi/180)\). That is, \(4.3 * (\pi/180) = 0.075\) radians.
02

Calculate the vertical height

After converting the slope, calculate the vertical height using sine function, as the height is the vertical component of the hypotenuse (which is the length of the road). Use the formula \(h = l*sin(\theta)\), where \(l\) is the length of the road, and \(\theta\) is the slope in radians. Substituting the given values, \(h = 1200*sin(0.075) = 90\) m.
03

Calculate potential energy

Having mass and height, now we can calculate the potential energy. The formula for gravitational potential energy is \(PE_{gravity} = m*g*h\), where \(m\) is the mass, \(g\) is the gravitational acceleration (which is \(9.8 m/s^2\) on Earth), and \(h\) is the height. Therefore, substituting these values: \(PE_{gravity} = 72*9.8*90 = 63504\) J.

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Key Concepts

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

Slope to Radians Conversion
Understanding the conversion of slope from degrees to radians is crucial when dealing with trigonometric functions, which are a fundamental component of various physics problems, including those involving gravitational potential energy. The slope given in degrees (\(4.3^\text{o}\)) represents the angle of the incline and translating it into radians allows for more straightforward calculations in subsequent steps.

To convert degrees to radians, you utilize the proportion that 180 degrees is equivalent to \(\pi\) radians. The conversion formula is \(\theta_\text{radians} = \theta_\text{degrees} \times (\pi/180)\). Applying this to our problem, the slope \(4.3^\text{o}\) becomes approximately 0.075 radians after the conversion. It's through this mathematical translation that one can then engage with trigonometric functions, which typically operate within the radian system in most scientific calculators.
Trigonometric Functions
Trigonometry is indispensable when it comes to analyzing forces and dimensions in physics, which often relate to angles. In the given scenario, we utilize the sine function鈥攐ne of the basic trigonometric functions鈥攖o determine the vertical height reached by the bike racer.

The sine function relates an angle of a right triangle to the ratio of the length of the opposite side (the vertical height, in our case) to the length of the hypotenuse (the length of the road). The formula \(h = l\times\sin(\theta)\) effectively gives us the height when we input the length of the slope (\(l\)) and the angle in radians (\(\theta\)). For our exercise, with a road length of 1200 m and an angle of 0.075 radians, the sine function calculates the vertical height to be 90 meters. This is essential in determining the change in the bike racer's gravitational potential energy.
Potential Energy Calculation
Moving on to the calculation of potential energy, we now focus on the energy stored due to an object's position within a gravitational field. This is known as gravitational potential energy (PE), and it is dependent on the object's mass, the height it is raised to, and the acceleration due to gravity (\(g\), which is \(9.8 m/s^2\) on Earth).

The formula for calculating gravitational potential energy is \(PE_\text{gravity} = m\times g\times h\), where \(m\) is the mass, \(g\) is the gravitational acceleration, and \(h\) is the height above the reference point. By substituting our determined variables (mass = 72 kg, gravitational acceleration = \(9.8 m/s^2\), and height = 90 m), the gravitational potential energy for the bike racer turns out to be 63504 Joules. This quantifies the racer鈥檚 increased potential energy throughout the climb, emphasizing the direct relationship between an object鈥檚 change in height and its potential energy in a gravitational field.

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Most popular questions from this chapter

A tennis ball bouncing on a hard surface compresses and then rebounds. The details of the rebound are specified in tennis regulations. Tennis balls, to be acceptable for tournament play, must have a mass of 57.5 g. When dropped from a height of \(2.5 \mathrm{m}\) onto a concrete surface, a ball must rebound to a height of \(1.4 \mathrm{m} .\) During impact, the ball compresses by approximately \(6 \mathrm{mm}\). How fast is the ball moving when it hits the concrete surface? (Ignore air resistance.) A. \(5 \mathrm{m} / \mathrm{s}\) B. \(7 \mathrm{m} / \mathrm{s}\) C. \(25 \mathrm{m} / \mathrm{s}\) D. \(50 \mathrm{m} / \mathrm{s}\)

\(\mathrm{A} 710 \mathrm{kg}\) car drives at a constant speed of \(23 \mathrm{m} / \mathrm{s}\). It is subject to a drag force of 500 N. What power is required from the car's engine to drive the car a. On level ground? b. Un a hill with a slone of \(2.0^{\circ} ?\)

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass \(m\) is fired into a block of mass \(M .\) The block, with the embedded bullet, then slides across a frictionless table and collides with a horizon- tal spring whose spring constant is \(k\). The opposite end of the spring is anchored to a wall. The spring's maximum compression \(d\) is measured. a. Find an expression for the bullet's initial speed \(v_{\mathrm{B}}\) in terms of \(m, M, k,\) and \(d\) Hint: This is a two-part problem. The bullet's collision with the block is an inelastic collision. What quantity is conserved in an inelastic collision? Subsequently the block hits a spring on a frictionless surface. What quantity is conserved in this collision? b. What was the speed of a \(5.0 \mathrm{g}\) bullet if the block's mass is \(2.0 \mathrm{kg}\) and if the spring, with \(k=50 \mathrm{N} / \mathrm{m},\) was compressed by \(10 \mathrm{cm} ?\) c. What fraction of the bullet's initial kinetic energy is "lost"? Where did it go?

An air-track glider undergoes a perfectly inelastic collision with an identical glider that is initially at rest. What fraction of the first glider's initial kinetic energy is transformed into thermal energy in this collision?

The spring in a retractable ballpoint pen is \(1.8 \mathrm{cm}\) long, with a \(300 \mathrm{N} / \mathrm{m}\) spring constant. When the pen is retracted, the spring is compressed by \(1.0 \mathrm{mm} .\) When you click the button to extend the pen, you compress the spring by an additional \(6.0 \mathrm{mm}\). How much energy is required to extend the pen?
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