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Q4.28(a):
Can you associate vectors with (a) the length of a wire bent into a loop,
Solution :
1. Theoretical Foundation: Defining a Vector
In classical mechanics and vector calculus, a quantity is classified as a vector if it possesses both a defined magnitude and a unique direction in space such that it obeys the parallelogram law of vector addition [Per the geometric axioms of vector spaces].
2. Analysis of the Physical Object: A Wire Loop
Consider a closed loop of wire $C$ residing in a plane. The length of the wire, $L$, is a scalar quantity representing the line integral of the arc length along the closed path: $L = \oint_C ds$.
To determine if this can be associated with a vector, we evaluate the vector area property. A closed loop does not possess a unique displacement vector because the starting point and ending point coincide (displacement $\Delta \vec{r} = 0$). However, a closed planar loop is uniquely associated with a vector area $\vec{A}$, defined as:
$\vec{A} = \oint_C \frac{1}{2} (\vec{r} \times d\vec{l})$
[By the application of Stokes' Theorem where the surface integral of the curl of a vector field is related to the line integral around its boundary].
3. Evaluation of "Length" as a Vector
- Magnitude: The length $L$ is a scalar value (e.g., $10\text{ cm}$). It is invariant under rotation and translation.
- Direction: A simple "length" of a wire has no inherent direction. If the wire is bent into an arbitrary shape, there is no single vector that can represent the "length" because the path of the wire changes at every point.
- Constraint: Associating a vector with the wire requires defining an orientation. While we can associate a vector area with the loop, we cannot associate the length of the wire with a vector.
4. Visual Representation of Vector Area vs. Scalar Length
5. Deductive Conclusion
The length of a wire bent into a loop is a scalar quantity. It represents the magnitude of the path taken, which does not have a specific direction associated with the path itself in a way that satisfies the properties of a vector (specifically, the length of the wire remains the same regardless of the orientation or shape of the loop in 3D space).
However, the closed loop itself can be associated with a vector area $\vec{A}$, where the magnitude is the area enclosed and the direction is the normal to the plane of the loop [Per the Right-Hand Thumb Rule for planar surfaces].
Final Solution: No, the length of a wire bent into a loop is a scalar quantity; it cannot be associated with a vector as it lacks a unique direction inherent to the measurement of its length.
More Questions from Class 11 Physics Motion in a Plane Additional Exercises
- 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.
- 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(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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