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Q4.26:
A vector has magnitude and direction. Does it have a location in space ? Can it vary with time ? Will two equal vectors a and b at different locations in space necessarily have identical physical effects ? Give examples in support of your answer.
Solution :
1. Conceptual Analysis of Vector Location
A vector is defined by its magnitude and direction. In mathematics, a "free vector" is defined purely by these two parameters, meaning its location in space is not fixed. However, in physical applications, vectors are categorized into two types:
- Free Vectors: These vectors can be shifted parallel to themselves without changing their physical significance (e.g., the velocity of a particle).
- Localized Vectors: These vectors have a specific point of application. Their physical effect is dependent on their location in space (e.g., force acting on a rigid body).
[Per the definition of vector fields and rigid body dynamics]
2. Temporal Variance of Vectors
A vector can indeed vary with time. A vector $\vec{A}(t)$ varies with time if either its magnitude or its direction (or both) changes as time $t$ progresses. Mathematically, this is expressed as $\frac{d\vec{A}}{dt} \neq 0$.
Example: A particle moving in a circular path with uniform speed has a constant magnitude of velocity $|\vec{v}|$, but the direction changes continuously. Thus, the velocity vector $\vec{v}(t)$ is time-dependent. [By the laws of Kinematics]
3. Physical Effects of Vectors at Different Locations
Two equal vectors $\vec{a}$ and $\vec{b}$ (where $|\vec{a}| = |\vec{b}|$ and direction of $\vec{a} = $ direction of $\vec{b}$) do not necessarily have identical physical effects if they are at different locations.
4. Comparative Examples
| Scenario | Physical Effect | Explanation |
|---|---|---|
| Force on a rigid body | Different effects | A force applied at the center of mass causes linear acceleration; the same force applied at a distance from the center causes rotation (torque). [Per the Principle of Moments] |
| Uniform Electric Field | Identical effects | A charge $q$ placed at any point in a uniform field $\vec{E}$ experiences the same force $\vec{F} = q\vec{E}$. [Per Coulomb’s Law] |
Final Solution: A vector is not inherently tied to a location in space (as in free vectors), but physical effects depend on location for localized vectors (e.g., torque). Vectors can vary with time if their magnitude or direction changes. Two equal vectors at different locations do not necessarily produce the same physical effect, as evidenced by the differing rotational outcomes of forces applied to different points on a rigid body.
More Questions from Class 11 Physics Motion in a Plane Additional Exercises
- Q4.27: A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector ? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector ?
- Q4.28(a): Can you associate vectors with (a) the length of a wire bent into a loop,
- Q4.28(b): Can you associate vectors with (b) a plane area,
- Q4.28(c): Can you associate vectors with (c) a sphere ? Explain.
- Q4.29: A bullet fired at an angle of $30^{\circ}$ with the horizontal hits the ground $3.0 km$ away. By adjusting its angle of projection, can one hope to hit a target $5.0 km$ away ? Assume the muzzle speed to be fixed, and neglect air resistance.
- Q4.30: A fighter plane flying horizontally at an altitude of $1.5 km$ with speed $720 km/h$ passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed $600 m s^{-1}$ to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit ? (Take $g = 10 m s^{-2}$ ).
- Q4.31: A cyclist is riding with a speed of $27 km/h$. As he approaches a circular turn on the road of radius $80 m$, he applies brakes and reduces his speed at the constant rate of $0.50 m/s$ every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?
- Q4.32(a): (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by $\theta(t) = \tan^{-1}\left(\frac{v_{0y} - gt}{v_{ox}}\right)$
- Q4.32(b): (b) Shows that the projection angle $\theta_0$ for a projectile launched from the origin is given by $\theta_0 = \tan^{-1}\left(\frac{4h_m}{R}\right)$ where the symbols have their usual meaning.
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