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Chapter 3: Problem 7

During a brake test, the rear-engine car is stopped from an initial speed of \(100 \mathrm{km} / \mathrm{h}\) in a distance of \(50 \mathrm{m} .\) If it is known that all four wheels contribute equally to the braking force, determine the braking force \(F\) at each wheel. Assume a constant deceleration for the 1500 -kg car.

Short Answer

Expert verified
The braking force for each wheel is approximately 2896.65 N.

Step by step solution

01

Convert Initial Speed to m/s

First, convert the initial speed from kilometers per hour to meters per second. The initial speed is given as 100 km/h. Use the conversion factor \(1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}\):\[ v_i = 100 \times \frac{5}{18} = \frac{500}{18} \approx 27.78 \, \text{m/s} \]
02

Apply the Equation of Motion to Find Deceleration

Use the kinematic equation to find deceleration \(a\):\[ v_f^2 = v_i^2 + 2a d \]Here, \(v_f = 0 \) (the car stops), \(v_i = 27.78 \, \text{m/s}\), and \(d = 50 \, \text{m}\):\[ 0 = (27.78)^2 + 2a \times 50 \]\[ 0 = 772.44 + 100a \]\[ a = -\frac{772.44}{100} \]\[ a = -7.7244 \, \text{m/s}^2 \]
03

Calculate Total Braking Force

Use Newton's second law to calculate the total braking force \(F_{\text{total}}\):\[ F_{\text{total}} = m \cdot a \]where \(m = 1500 \, \text{kg}\) is the mass of the car and \(a = -7.7244 \, \text{m/s}^2\) is the acceleration:\[ F_{\text{total}} = 1500 \times 7.7244 = 11586.6 \, \text{N} \]
04

Determine Braking Force per Wheel

Since all four wheels contribute equally to the braking force, divide the total braking force by 4 to find the force on each wheel:\[ F_{\text{wheel}} = \frac{F_{\text{total}}}{4} = \frac{11586.6}{4} = 2896.65 \, \text{N} \]
05

Final Result

Each wheel contributes a braking force of approximately 2896.65 N.

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Key Concepts

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

Braking Force Calculation
Braking force is the force applied to stop a moving vehicle. It is crucial for determining how quickly a vehicle can stop. In the context of this exercise, we calculate the braking force using the principles of engineering mechanics. The key point is that the total braking force acts through all four wheels equally. To find it, we first determine the total force required to bring the car to a halt and then divide by the number of wheels. Here is a straightforward breakdown of the steps:
  • Calculate the deceleration needed to stop the car.
  • Use the car's mass to find the total force exerted by the brakes.
  • Divide the total force by four, since each of the car's wheels contributes equally.
By doing this, we ensure that each wheel applies the same amount of force required to safely and effectively stop the car.
Kinematics
Kinematics is the branch of mechanics dealing with the motion of objects. It involves understanding how objects move without considering the forces that cause the motion. Here, we use kinematic equations to find the deceleration of the car.To solve this, we apply the kinematic equation:\[ v_f^2 = v_i^2 + 2a d \]where:- \( v_f \) is the final speed (0, since the car stops).- \( v_i \) is the initial speed, converted to meters per second.- \( a \) is the acceleration (which is a negative value here, indicating deceleration).- \( d \) is the stopping distance.By rearranging the equation, we calculate the deceleration needed to stop the car. Understanding this process is essential in solving problems involving motion and stopping distances in engineering mechanics.
Newton's Second Law
Newton's second law of motion is pivotal in calculating forces. According to this law, the force exerted on an object equals the mass of the object multiplied by its acceleration: \( F = m imes a \). In our exercise, this principle helps us determine the total braking force.Here are the steps:
  • We already calculated the deceleration, which in terms of force, acts as the negative acceleration required to stop the car.
  • The car's mass is given as 1500 kg.
  • Using the formula, multiply the mass by the deceleration to find the total braking force.
Understanding this relationship allows engineers to predict how a vehicle will respond to forces like friction and braking, ensuring both safety and efficiency in vehicle design.
Deceleration
Deceleration is the rate at which an object slows down. It is essentially negative acceleration and a crucial parameter in designing a car's braking system. To calculate deceleration in our braking scenario, we used the kinematic equation. Key steps in understanding deceleration include:
  • Convert the vehicle's initial speed to meters per second for consistency in units.
  • Recognize deceleration as a negative change in velocity over time, here stressing over a known distance rather than time.
  • Apply the kinematic formula to solve for the deceleration value.
This concept of deceleration is fundamental in engineering, ensuring vehicles can adequately stop within safe distances under varying conditions.

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Most popular questions from this chapter

The standard test to determine the maximum lateral acceleration of a car is to drive it around a 200 -ft-diameter circle painted on a level asphalt surface. The driver slowly increases the vehicle speed until he is no longer able to keep both wheel pairs straddling the line. If this maximum speed is \(35 \mathrm{mi} / \mathrm{hr}\) for a \(3000-\mathrm{lb}\) car, determine its lateral acceleration capability \(a_{n}\) in \(g\) 's and compute the magnitude \(F\) of the total friction force exerted by the pavement on the car tires.

A 60 -kg woman holds a 9 -kg package as she stands within an elevator which briefly accelerates upward at a rate of \(g / 4 .\) Determine the force \(R\) which the elevator floor exerts on her feet and the lifting force \(L\) which she exerts on the package during the acceleration interval. If the elevator support cables suddenly and completely fail, what values would \(R\) and \(L\) acquire?

Car \(A\) weighing 3200 lb and traveling north at \(20 \mathrm{mi} / \mathrm{hr}\) collides with car \(B\) weighing \(3600 \mathrm{lb}\) and traveling at \(30 \mathrm{mi} / \mathrm{hr}\) as shown. If the two cars become entangled and move together as a unit after the crash, compute the magnitude \(v\) of their common velocity immediately after the impact and the angle \(\theta\) made by the velocity vector with the north direction.

The uniform slender rod of length \(L,\) mass \(m,\) and cross-sectional area \(A\) is rotating in a horizontal plane about the vertical central axis \(0-0\) at a constant high angular velocity \(\omega .\) By analyzing the horizontal forces on the accelerating differential element shown, derive an expression for the tensile stress \(\sigma\) in the rod as a function of \(r .\) The stress, commonly referred to as centrifugal stress, equals the tensile force divided by the cross-sectional area \(A\).

The 600,000 -lb jet airliner has a touchdown velocity \(v=120 \mathrm{mi} / \mathrm{hr}\) directed \(\theta=0.5^{\circ}\) below the horizontal. The touchdown process of the eight main wheels takes 0.6 sec to complete. Treat the aircraft as a particle and estimate the average normal reaction force at each wheel during this 0.6 -sec process, during which tires deflect, struts compress, etc. Assume that the aircraft lift equals the aircraft weight during the touchdown.
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