Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 95

Chapter 8, Problem 95

Proper design of automobile braking systems must account for heat buildup under heavy braking. Calculate the thermal energy dissipated from brakes in a 1500-kg car that descends a 17° hill. The car begins braking when its speed is 95 km/h and slows to a speed of 35 km/h in a distance of 0.30 km measured along the road.

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Convert the initial and final speeds of the car from km/h to m/s using the formula: \( v_{m/s} = \frac{v_{km/h} \times 1000}{3600} \).

Calculate the change in kinetic energy of the car using the formula: \( \Delta KE = \frac{1}{2} m (v_f^2 - v_i^2) \), where \( m \) is the mass of the car, \( v_i \) is the initial velocity, and \( v_f \) is the final velocity.

Determine the work done against gravity as the car descends the hill. First, calculate the vertical height \( h \) using the relationship \( h = d \sin(\theta) \), where \( d \) is the distance traveled along the road and \( \theta \) is the angle of the hill. Then, calculate the gravitational potential energy change using \( \Delta PE = m g h \), where \( g \) is the acceleration due to gravity.

Add the change in kinetic energy (\( \Delta KE \)) and the change in gravitational potential energy (\( \Delta PE \)) to find the total thermal energy dissipated by the brakes: \( Q = \Delta KE + \Delta PE \).

Ensure all units are consistent throughout the calculations (e.g., mass in kg, velocity in m/s, distance in meters, and energy in joules) to avoid errors in the final result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. In this scenario, the car's initial and final speeds will determine the change in kinetic energy as it brakes, which directly relates to the thermal energy dissipated in the braking system.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In the context of braking, the work done by the brakes to slow the car down converts kinetic energy into thermal energy, which is dissipated as heat. This principle is essential for calculating the energy lost during the braking process.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another, typically occurring through conduction, convection, or radiation. In braking systems, the heat generated from friction between brake pads and rotors must be managed to prevent overheating, making it crucial to understand how thermal energy dissipates during braking.

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Related Practice

Textbook Question

Suppose we have three masses, m₁ , m₂ and m₃, that initially are extremely (≈ infinitely) far apart from each other. The work needed to bring them to the positions shown in Fig. 8–50 is W = - G ((m₁m₂/ r₁₂) + (m₁m₃/r₁₃) + (m₂m₃/r₂₃)). Is W equal to the binding energy of the system—that is, is W equal to the energy required to separate the components by an infinite distance? Explain.

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Water flows slowly over a dam at the rate of 320 kg/s and falls vertically 88 m before striking the turbine blades. Calculate the rate at which mechanical energy is transferred to the turbine blades, assuming 55% efficiency.

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Textbook Question

The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential) U(r) = -(a/r⁶) + (b/r¹²), where r is the distance between the two atoms and a and b are positive constants. At what values of r is U(r) a minimum? A maximum?

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How much work can a 3.0-hp motor do in 1.0h?

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Textbook Question

The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or F(r) = F₀ [2(σ/r)¹³ - (σ/r)⁷], where r is the separation between the two atoms, and σ and F₀ are constants. For an oxygen molecule (which is diatomic) F₀ = 9.60 x 10⁻¹¹ N and σ = 3.50 x 100⁻¹¹ m. Integrate the equation for F(r) to determine the potential energy U(r) of the oxygen molecule.

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Textbook Question

If you stand on a bathroom scale, the spring inside the scale compresses 0.60 mm, and it tells you your weight is 760 N. Now if you jump on the scale from a height of 1.0 m, what does the scale read at its peak? Assume Hooke’s law holds.

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