/*! 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 6 A force of constant magnitude \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 6

A force of constant magnitude \(F=10 \mathrm{~N}\) acts on a particle moving in a plane such that it is perpendicular to the velocity \(\vec{v}(|\vec{v}|=v=5 \mathrm{~m} / \mathrm{s})\) of the body, and the force is always directed towards a fixed point. Then the angle (in radian) turned by the velocity vector of the particle as it covers a distance \(S=10 \mathrm{~m}\) is (take: mass of the particle as \(m=2 \mathrm{~kg}\) )

Short Answer

Expert verified
The velocity vector turns by 2 radians as the particle covers 10 m.

Step by step solution

01

Understanding the Motion

The problem describes a situation of uniform circular motion, as the force is always perpendicular to the velocity vector and directed towards a fixed point. This indicates that the particle is moving in a circular path.
02

Formula for Centripetal Force

For a body moving in a circular motion, the centripetal force is given by the formula: \( F_c = \frac{mv^2}{r} \), where \( m \) is mass, \( v \) is the velocity, and \( r \) is the radius of the trajectory.
03

Calculate the Radius of the Path

Since \( F = F_c \), we have: \( \frac{mv^2}{r} = 10 \). Substituting the given values: \( \frac{2 \cdot 5^2}{r} = 10 \rightarrow \frac{50}{r} = 10 \). Thus, solving for \( r \), we find: \( r = \frac{50}{10} = 5 \) meters.
04

Calculate the Angle in Radians

The arc length \( S \) for the angle \( \theta \) in a circle is given by the formula \( S = r\theta \). For \( S = 10 \) m and \( r = 5 \) m, substituting into the formula gives: \( 10 = 5\theta \). Solving for \( \theta \), we get \( \theta = \frac{10}{5} = 2 \) radians.

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 is a fundamental concept when studying objects in circular motion. It refers to the force that acts on an object moving in a circle, always pointing towards the center of the circle. This inward force is crucial because it keeps the object on its circular path. Without centripetal force, an object would continue moving in a straight line due to inertia.
- The formula for centripetal force is given by: \[ F_c = \frac{mv^2}{r} \] where:
  • \( F_c \) is the centripetal force
  • \( m \) is the mass of the object
  • \( v \) is the velocity of the object
  • \( r \) is the radius of the circular path
- Centripetal force is not a new type of force on its own. Instead, it is the name given to forces such as tension, gravity, or friction when they result in circular motion. For this exercise, the force acting on our particle is 10 N, which aligns perfectly with the force required to keep it in circular motion.
It's important to remember that without this constant centripetal force acting towards the center, the object's velocity vector would not change direction, and thus, it wouldn't move along a circular path.
Uniform Circular Motion
Uniform circular motion refers to the motion of an object traveling in a circle with a constant speed. Even though the speed remains constant, the velocity does not. This is because velocity is a vector, meaning it has both magnitude and direction, and in circular motion, the direction is continuously changing.
- In our problem, the particle exhibits uniform circular motion because:
  • The speed \( v \) is 5 m/s, and it remains constant.
  • The centripetal force continually redirects the particle towards the center.
- Since the speed is constant, calculating other quantities like the radius of its path is essential for understanding its motion.
Analyzing an object's uniform circular motion helps in predicting how the object moves, calculating forces, and understanding the interaction of different physical quantities. Uniform circular motion is significant because, despite the constant speed, the constant change in direction implies that the object is always accelerating.
Angle in Radians
Angles in circular motion are often measured in radians because it gives a direct relationship with the path the object travels over the circular path. Radians provide a straightforward way to relate the arc length of the circle to the radius, which makes calculating many related quantities easier and more intuitive.
- The formula to calculate the angle \( \theta \) in radians is: \[ S = r\theta \] where:
  • \( S \) is the arc length, which is the distance the object has traveled along the circular path.
  • \( r \) is the radius of the circular path.
  • \( \theta \) is the angle in radians.
- In our problem, the arc length \( S \) is 10 meters, and the radius \( r \) is 5 meters, leading to an angle \( \theta = 2 \) radians.
Understanding the angle in radians is fundamental in cases like this because it aids in describing how much of the circular path the object has covered. Unlike degrees, radians offer a more natural link to the other quantities at play in circular motion, making calculations more fluid and meaningful.

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

Two strings of length \(l=0.5 \mathrm{~m}\) each are connected to a block of mass \(m=2 \mathrm{~kg}\) at one end and their ends are attached to the point \(A\) and \(B, 0.5 \mathrm{~m}\) apart on a vertical pole which rotates with a constant angular velocity \(\omega=7 \mathrm{rad} / \mathrm{s}\). Find the ratio \(T_{1} / T_{2}\) of tension in the upper string \(\left(T_{1}\right)\) and the lower string \(\left(T_{2}\right)\). [Use \(\left.g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right]\)

The kinetic energy \(K\) of a particle moving along a circle of radius \(R\) depends upon the distance \(s\) as \(K=a s^{2} .\) The force acting on the particle is (1) \(2 a \frac{s^{2}}{R}\) (2) \(2 a s\left[1+\frac{s^{2}}{R^{2}}\right]^{1 / 2}\) (3) \(2 a s\) (4) \(2 a\)

A bead of mass \(m\) is free to slide on a fixed horizontal circular wire of radius \(R\). At time \(t=0\), it is given a velocity \(v_{0}\) along the tangent to the circle. If the coefficient of kinetic friction between the bead and the wire is \(\mu_{k}\). Then magnitude of tangential acceleration at \(t=0\) will be (1) \(\mu_{k} g\) (2) \(\mu_{k} \frac{v^{2}}{R}\) (3) \(\frac{\mu_{k}}{R} \sqrt{v^{2}+g^{2} R^{2}}\) (4) \(\mu_{k}\left[g+\frac{v^{2}}{R}\right]\)

A particle tied at one end of an inextensible string of length \(R\), whose another end is fixed. Particle is performing circular motion in vertical plane. Ratio of maximum to minimum speed is \(\lambda\). Choose the correct option(s). (1) Speed at the top most point is \(\sqrt{\left(\frac{4 \lambda^{2}}{\lambda^{2}-1}\right) g R}\) (2) Speed at the top most point is \(\sqrt{\left(\frac{4}{\lambda^{2}-1}\right) g R}\) (3) Ratio of maximum to minimum tension is \(\left(\frac{5 \lambda^{2}-1}{5-\lambda^{2}}\right)\) (4) Ratio of tension in string to weight when string is horizontal is \(2\left(\frac{\lambda^{2}-1}{\lambda^{2}+1}\right)\)

Two bodies of masses \(m\) and \(4 m\) are attached to a light string as shown in figure. A body of mass \(m\) hanging from string is executing oscillations with angular amplitude \(60^{\circ}\), while other body is at rest on a horizontal surface. The minimum coefficient of friction between mass \(4 m\) and the horizontal surface is (here pulley is light and smooth) (1) \(\frac{1}{4}\) (2) \(\frac{3}{4}\) (3) \(\frac{1}{2}\) (4) \(\frac{1}{8}\)
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Turning Points in Physics

Read Explanation

Kinematics Physics

Read Explanation

Mechanics and Materials

Read Explanation

Wave Optics

Read Explanation

Oscillations

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.