/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Three identical cars \(A, B\) an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 21

Three identical cars \(A, B\) and \(C\) are moving at the same speed on three bridges. The car \(A\) goes on a plane bridge, \(B\) on a bridge convex upwards and \(C\) goes on a bridge concave upwards. Let \(F_{A}, F_{B}\) and \(F_{C}\) be the normal forces exerted by the cars on the bridges when they are at the middle of the bridges. Then: (a) \(F_{A}\) is maximum of the three forces (b) \(F_{B}\) is maximum of the three forces (c) \(F_{C}\) is maximum of the three forces (d) \(F_{A}=F_{B}=F_{C}\)

Short Answer

Expert verified
\( F_{C} \) is maximum of the three forces.

Step by step solution

01

Understand the Problem

We have three identical cars moving with the same speed on three different types of bridges. Car A is on a flat bridge, car B is on a convex bridge (curving upward), and car C is on a concave bridge (curving downward). We need to find the relation between the normal forces exerted by each car on the bridge at the midpoint of the bridge.
02

Analyze the Forces on Car A

Since car A is moving on a flat bridge, the only forces acting on it are its weight due to gravity \(mg\) downward and the normal force \(F_{A}\) exerted by the bridge upward. Thus, the normal force \(F_{A}\) is equal to the gravitational force, \(F_{A} = mg\).
03

Analyze the Forces on Car B

Car B is moving on a convex bridge. As it moves to the top of the bridge, its motion adds an outward centripetal force. This subtracts from the normal force, resulting in a reduced normal force as compared to car A. Hence, \(F_{B} < mg\).
04

Analyze the Forces on Car C

Car C is moving on a concave bridge. As it travels, the centripetal acceleration is directed inward, which implies the bridge must exert an additional normal force to maintain the curvature way downwards. This results in an increased normal force on car C, \(F_{C} > mg\).
05

Compare the Normal Forces

Based on the above analysis, since \(F_{A} = mg\), \(F_{B} < mg\), and \(F_{C} > mg\), it follows that \(F_{C} > F_{A} > F_{B}\). Thus, the normal force is greatest for car C.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Centripetal Force
Centripetal force plays a crucial role when an object moves along a curved path. It is the force that keeps an object moving in a circular motion. This force is directed towards the center of the curvature.
For a car traveling around a curve, the centripetal force is necessary to change the direction of its velocity towards the center.
  • For a car on a convex bridge (curved upwards), the outward centripetal force acts in the same direction as gravity. This reduces the normal force exerted by the bridge.
  • For a car on a concave bridge (curved downwards), the centripetal force acts opposite to gravity. This increases the normal force needed to keep the car moving along the curve.
Understanding this helps us to see how different curves affect the forces on a moving car.
Gravitational Force
Gravitational force is the force by which a planet, like Earth, attracts an object towards its center. This force is always acting downwards and is calculated by the formula\[ F_g = mg \]where:
  • \(m\) is the mass of the object
  • \(g\) is the acceleration due to gravity, approximately \(9.8 \text{m/s}^2\) on Earth
In the context of the exercise, gravity constantly pulls the car downward towards Earth's center.
It acts on all cars equally but interacts differently depending on the bridge's shape.
On flat ground, gravity directly equals the normal force. On curved surfaces, the normal force will differ due to additional centripetal forces.
Curved Bridges
The shape of a bridge impacts how forces act on vehicles moving over it.
There are two primary shapes being considered - convex (curving upwards) and concave (curving downwards):
  • **Convex Bridges**: As a car moves over a convex up bridge, the curved path causes the normal force to decrease. This happens because part of the gravitational force is used to provide centripetal force for the car's circular motion, reducing the normal force.
  • **Concave Bridges**: On a concave down bridge, a car experiences a stronger normal force because the curve requires additional force to maintain motion inwards, counteracting gravity with both the normal force and an inward centripetal force.
Choosing bridge shapes in road design comes down to engineering decisions based on vehicle dynamics and safety.
Car Dynamics
Car dynamics involve understanding how forces affect the motion of a car.
These dynamics are particularly intriguing when considering a car moving over a bridge:
  • Speed, mass, and the shape of the path determine the centripetal and gravitational forces on the car.
  • The normal force, as a result, will fluctuate based on these factors and the bridge's curvature.
  • Flat bridges offer a standard scenario where normal force equals gravitational force for balanced dynamics.
On curved bridges, however, the dynamics become more complex as additional forces modify how vehicles interact with the surface.
This dictates not only safety considerations but also informs engineering decisions on bridge design and vehicle stability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The angular displacement of the rod is defined as \(\theta=\frac{3}{20} t^{2}\) where \(\theta\) is in radian and \(t\) is in second. The collar \(B\) slides along the rod in such a way that its distance from \(O\) is, \(r=0.9-0.12 t^{2}\) where \(r\) is in metre and \(t\) is in second. The velocity of collar at \(\theta=30^{\circ}\) is: (a) \(0.45 \mathrm{~m} / \mathrm{s}\) (b) \(0.48 \mathrm{~m} / \mathrm{s}\) (c) \(0.52 \mathrm{~m} / \mathrm{s}\) (d) \(0.27 \mathrm{~m} / \mathrm{s}\)

Particles are released from rest at \(A\) and slide down the smooth surface of height \(h\) to a conveyor \(B\). The correct angular velocity \(\omega\) of the conveyor pulley of radius \(r\) to prevent any sliding on the belt as the particles transfer to the conveyor is: (a) \(\sqrt{2 g h}\) (b) \(\frac{2 g h}{r}\) (c) \(\frac{\sqrt{2 g h}}{r}\) (d) \(\frac{2 g h^{2}}{r^{2}}\)

A person wants to drive on the vertical surface of a large cylindrical wooden well commonly known as death well in a circus. The radius of well is \(R\) and the coefficient of friction between the tyres of the motorcycle and the wall of the well is \(\mu_{s}\). The minimum speed, the motorcyclist must have in order to prevent slipping should be : (a) \(\sqrt{\frac{R_{g}}{\mu_{\mathrm{s}}}} \quad\) (b) \(\sqrt{\frac{\mu_{\mathrm{s}}}{\mathrm{Rg}}}\) (c) \(\sqrt{\frac{\mu_{i} g}{R}}\) (d) \(\sqrt{\frac{R}{\mu_{s}}}\)

Mark correct option or options from the following : (a) In the case of circular motion of a particle, centripetal force may be balanced by centrifugal force (b) In the non-inertial reference frame centrifugal force is real force (c) In the inertial reference frame, centrifugal force is real force (d) Centrifugal force is always pseudo force

The small spherical balls are free to move on the inner surface of the rotating spherical chamber of radius \(R=0.2 \mathrm{~m} .\) If the balls reach a steady state at angular position \(\theta=45^{\circ}\), the angular speed \(\omega\) of device is: (a) \(8 \mathrm{rad} / \mathrm{sec}\) (b) \(2 \mathrm{rad} / \mathrm{sec}\) (c) \(3.64 \mathrm{rad} / \mathrm{sec}\) (d) \(9.34 \mathrm{rad} / \mathrm{sec}\)
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Energy Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Mechanics and Materials

Read Explanation

Translational Dynamics

Read Explanation

Turning Points in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.