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Chapter 4: Problem 46

A particle is found to be at rest when seen from frame \(S_{1}\) and moving with a constant velocity when seen from another frame \(S_{2}\). Mark the possible points from the following : (a) both the frames are inertial (b) both the frames are non-inertial (c) \(S_{1}\) is non-inertial and \(S_{2}\) is inertial (d) both (a) and (b) are correct

Short Answer

Expert verified
Both (a) and (b) are correct.

Step by step solution

01

Understanding the problem

We need to figure out the relationship between the frames of reference \(S_1\) and \(S_2\) and whether they are inertial or non-inertial, based on the motion of the particle as observed in each frame. In one frame, the particle is at rest, and in the other, it is moving at a constant velocity.
02

Define inertial frames

In physics, an inertial frame is a frame of reference in which a body not subjected to forces moves at a constant velocity (which could be zero), or it remains at rest. Hence, if either frame (\(S_1\) or \(S_2\)) shows the particle at constant velocity, it could be inertial.
03

Analyzing rest in frame \(S_1\)

A particle being at rest in frame \(S_1\) could mean that \(S_1\) is an inertial frame with zero velocity or a non-inertial frame. Further analysis is needed to determine this.
04

Analyzing constant velocity in frame \(S_2\)

A particle moving at constant velocity in frame \(S_2\) indicates that \(S_2\) is likely an inertial frame, as objects in inertial frames continue in uniform motion when not acted upon by a net external force.
05

Conclusion on frame types

Given that a particle is at rest in \(S_1\) (which could be inertial or non-inertial) and moves with constant velocity in \(S_2\) (suggesting \(S_2\) is inertial), both frames could technically be inertial: One showing zero velocity (rest), and one showing constant velocity (movement). Alternatively, both could be non-inertial if external forces explain both observations equally without maintaining uniform motion as per Newton's first law.

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Key Concepts

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

Inertial Frames
Inertial frames are an important concept in physics, especially in the study of motion. They are frames of reference where objects move at a constant velocity unless acted upon by an external force. This aligns with Newton's first law of motion.

In simple terms, think of an inertial frame as being a peaceful environment where no unbalanced forces exist. For example:
  • A spaceship drifting in the vacuum of space without its engines on.
  • A train moving at a steady speed without changing its direction or speed abruptly.
If a particle appears at rest in such a frame, it could mean the frame itself is stationary or moving at a constant velocity. Understanding this is key in identifying the nature of reference frames, such as in our exercise with frames \(S_1\) and \(S_2\).
Non-Inertial Frames
Non-inertial frames are quite different from inertial frames. These are reference frames that are accelerating. In a non-inertial frame, you may feel forces acting upon you even when sitting still.

Exploring examples can help clarify:
  • Imagine sitting in a car that suddenly accelerates. You feel pushed back into your seat.
  • Or standing in an elevator that starts descending quickly, making you feel lighter.
These frames do not obey Newton's first law because there are apparent forces 鈥� often termed "fictitious forces" 鈥� that arise due to the frame's acceleration. In understanding particle motion within frame \(S_1\) and \(S_2\), acknowledging whether the frames could be non-inertial frames helps fully explore all possible scenarios.
Constant Velocity
Constant velocity is crucial in distinguishing whether a frame is inertial. It implies that an object's speed and direction are unchanging.

This term becomes particularly handy when considering motion from different perspectives:
  • Within any inertial frame, the absence of net forces allows for constant velocity motion. This could be complete stillness or movement without speed or direction change.
  • A constant velocity observed in frame \(S_2\) suggests it behaves like an inertial frame, as Newton's laws aptly describe the motion.
Recognizing constant velocity aids in deducing the characteristics of the reference frames involved, especially as evaluated in our exercise.
Particle Motion
Particle motion is the study of how a particle changes position in relation to a frame of reference. This can vary significantly based on whether the frame is inertial or non-inertial.

Understanding how motion is perceived can be influenced by reference frames:
  • For \(S_1\), seeing the particle at rest can hint at zero motion or a balance of forces if it鈥檚 an inertial frame.
  • If both \(S_1\) and \(S_2\) provide observations that comply with these definitions, these frames can tell us a lot about the underlying forces, or lack thereof, acting on the particle.
  • Hence, examining particle motion is essential for correctly interpreting the inertial or non-inertial nature of reference frames.
Ultimately, how a particle behaves in various frames of reference offers insight into fundamental physical principles, aiding in problem-solving endeavors like our exercise.

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Most popular questions from this chapter

Let \(F, F_{N}\) and \(f\) denote the magnitudes of the contact force, normal force and the frictional force exerted by one surface on the other kept in contact. If one of these is zero, then : (a) \(F>F_{N}\) (b) \(F>f\) (c) \(F_{N}-f

Mark correct option or options: (a) The normal reaction and gravitational force on a body placed on a surface are action-reaction pair (b) The normal reaction on a body placed on a rough surface is always equal to weight of the body (c) \(v^{2}=u^{2}+2 g h\) is always applicable to a falling body on the earth in the absence of air (d) All are wrong

A particle of mass \(m\) moves on the \(x\) -axis under the influence of a force of attraction towards the origin \(O\) given by \(F=-\frac{k}{x} \vec{i} .\) If the particle starts from rest at \(x=a\). The speed it will attain to reach at distance \(x\) from the origin \(O\) will be : (a) \(\sqrt\left[\sqrt{\frac{2 k}{m}}\left[\frac{x-a}{a x}\right]^{1 / 2}\right.\) (b) \(\sqrt{\frac{2 \hbar}{m}}\left[\frac{a+k}{a x}\right]^{1 / 2}\) (c) \(\sqrt{\frac{k}{m}}\left[\frac{a x}{x-a}\right]\) (d) \(\sqrt{\frac{m}{2 k}}\left[\frac{a-x}{a x}\right]^{1 / 2}\)

A particle of mass \(m\) moves on the \(x\) -axis as follows. It starts from rest at \(t=0\) from the point \(x=0\), and comes to rest at \(t=1\) at the point \(x=1\). No other information is available about its motion at intermediate time \((04\) at point or some points in its path (d) both (a) and (c) are correct

A \(0.1 \mathrm{~kg}\) body moves at a constant speed of \(10 \mathrm{~m} / \mathrm{s}\). It is pushed by applying a constant force for \(2 \mathrm{sec}\). Due to this force, it starts moving exactly in the opposite direction with a speed of \(4 \mathrm{~m} / \mathrm{s}\). Then: (a) the deceleration of the body is \(7 \mathrm{~m} / \mathrm{s}^{2}\) (b) the magnitude of change in momentum is \(1.4 \mathrm{~kg}-\mathrm{m} / \mathrm{sec}\) (c) impulse of the force is \(1.4 \mathrm{Ns}\) (d) the force which acts on the ball is \(0.7 \mathrm{~N}\) (e) all of the above
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