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Q4.24(e):
Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
Solution :
Step 1: Theoretical Definition of a Scalar Quantity
A physical quantity is defined as a scalar if it is invariant under a transformation of the coordinate system. Mathematically, if $S$ is a scalar quantity in a coordinate system $K$ with axes $(x, y, z)$, and $S'$ is the same quantity in a coordinate system $K'$ with axes $(x', y', z')$ obtained by rotating the axes, then $S = S'$ must hold true [Per the definition of rotational invariance].
Step 2: Analysis of Coordinate Orientation
The orientation of axes refers to the rotation of the basis vectors $\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}$ to new basis vectors $\mathbf{\hat{i}'}, \mathbf{\hat{j}'}, \mathbf{\hat{k}'}$ through an orthogonal transformation matrix $R$, where $R^T R = I$.
For a scalar quantity $S$, its magnitude is independent of the spatial direction of the observer's chosen coordinate frame. Unlike vectors, which transform according to $\mathbf{A}' = R\mathbf{A}$, a scalar remains a constant value regardless of the orientation of the observer.
Step 3: Verification with Physical Examples
Consider the following examples to validate the statement:
| Quantity | Reasoning |
|---|---|
| Mass ($m$) | The mass of a particle is an intrinsic property. Rotating the $x, y, z$ axes does not change the amount of matter in the body. |
| Temperature ($T$) | Temperature is a thermodynamic state function. An observer looking at a thermometer from any orientation will measure the same scalar value. |
| Speed ($v$) | While velocity ($\mathbf{v}$) is a vector that changes components under rotation, the speed ($v = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$) is invariant [By the property of Orthogonal Transformation]. |
Step 4: Geometric Illustration of Scalar Invariance
Step 5: Conclusion on Validity
The statement is True. Because scalar quantities represent physical magnitudes that do not possess directional components, they do not transform when the observer rotates their coordinate system. The scalar product (dot product) of two vectors, $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta$, is a primary example of a scalar that remains constant because both the magnitudes and the angle $\theta$ between vectors are invariant under coordinate rotation.
Final Solution: The statement is True; a scalar quantity is invariant under any rotation of the coordinate axes because it possesses magnitude only, lacking the directional dependence that necessitates vector transformation.
More Questions from Class 11 Physics Motion in a Plane EXERCISES
- Q3.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q3.10: On an open ground, a motorist follows a track that turns to his left by an angle of (60^{\circ}) after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q3.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q3.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q3.12: The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of (40 m s^{-1}) can go without hitting the ceiling of the hall ?
- Q3.13: A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ?
- Q3.14: A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone ?
- Q3.15: An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
- Q3.16(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q3.16(b): Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
- Q3.16(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q3.17(a): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find the (v) and (a) of the particle?
- Q3.17(b): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (b) What is the magnitude and direction of velocity of the particle at (t = 2.0 s) ?
- Q3.18(a): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
- Q3.18(b): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (b) What is the speed of the particle at the time ?
- Q3.19: (\hat{i}) and (\hat{j}) are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors (\hat{i} + \hat{j}), and (\hat{i} - \hat{j})? What are the components of a vector (A= 2 \hat{i} + 3 \hat{j}) along the directions of (\hat{i} + \hat{j}) and (\hat{i} - \hat{j})? [You may use graphical method]
- Q3.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q3.20(a): For any arbitrary motion in space, which of the following relations are true : (a) (v_{average} = (\frac{1}{2}) (v(t_{1}) + v(t_{2})))
- Q3.20(b): For any arbitrary motion in space, which of the following relations are true : (b) (v_{average} = \frac{[r(t_{2}) - r(t_{1})]}{t_{2} – t_{1}})
- Q3.20(c): For any arbitrary motion in space, which of the following relations are true : (c) (v(t) = v(0) + a t)
- Q3.20(d): For any arbitrary motion in space, which of the following relations are true : (d) (r(t) = r(0) + v(0)t + (\frac{1}{2}) a t^{2})
- Q3.20(e): For any arbitrary motion in space, which of the following relations are true : (e) (a_{average} = \frac{[v(t_{2}) - v(t_{1})]}{t_{2} – t_{1}}) (The ‘average’ stands for average of the quantity over the time interval (t_{1}) to (t_{2}))
- Q3.21(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q3.21(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q3.21(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q3.21(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q3.21(e): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
- Q3.22: An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s a part is (30^{\circ}), what is the speed of the aircraft ?
- Q3.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q3.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q3.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q3.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q3.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q3.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q3.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q3.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q3.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q3.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q3.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q3.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q3.6(a): Establish the following vector inequalities geometrically or otherwise : (a) (|a+b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(b): Establish the following vector inequalities geometrically or otherwise : (b) (|a+b| > ||a| −|b||) When does the equality sign above apply?
- Q3.6(c): Establish the following vector inequalities geometrically or otherwise : (c) (|a−b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(d): Establish the following vector inequalities geometrically or otherwise : (d) (|a−b| > ||a| − |b||) When does the equality sign above apply?
- Q3.7(a): Given (a + b + c + d = 0), which of the following statements are correct : (a) (a), (b), (c), and (d) must each be a null vector,
- Q3.7(b): Given (a + b + c + d = 0), which of the following statements are correct : (b) The magnitude of ((a + c)) equals the magnitude of ((b + d)),
- Q3.7(c): Given (a + b + c + d = 0), which of the following statements are correct : (c) The magnitude of (a) can never be greater than the sum of the magnitudes of (b), (c), and (d),
- Q3.7(d): Given (a + b + c + d = 0), which of the following statements are correct : (d) (b + c) must lie in the plane of (a) and (d) if (a) and (d) are not collinear, and in the line of (a) and (d), if they are collinear ?
- Q3.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 3.19. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q3.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (a) net displacement,
- Q3.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (b) average velocity,
- Q3.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
- Q4.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q4.10: On an open ground, a motorist follows a track that turns to his left by an angle of $60^{\circ}$ after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q4.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q4.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q4.12: Rain is falling vertically with a speed of $30 m s^{-1}$. A woman rides a bicycle with a speed of $10 m s^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella ?
- Q4.13: A man can swim with a speed of $4.0 km/h$ in still water. How long does he take to cross a river $1.0 km$ wide if the river flows steadily at $3.0 km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?
- Q4.14: In a harbour, wind is blowing at the speed of $72 km/h$ and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of $51 km/h$ to the north, what is the direction of the flag on the mast of the boat ?
- Q4.15: The ceiling of a long hall is $25 m$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40 m s^{-1}$ can go without hitting the ceiling of the hall ?
- Q4.16: A cricketer can throw a ball to a maximum horizontal distance of $100 m$. How much high above the ground can the cricketer throw the same ball ?
- Q4.17: A stone tied to the end of a string $80 cm$ long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in $25 s$, what is the magnitude and direction of acceleration of the stone ?
- Q4.18: An aircraft executes a horizontal loop of radius $1.00 km$ with a steady speed of $900 km/h$. Compare its centripetal acceleration with the acceleration due to gravity.
- Q4.19(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q4.19(b): Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
- Q4.19(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q4.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q4.20(a): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (a) Find the $\boldsymbol{v}$ and $\boldsymbol{a}$ of the particle?
- Q4.20(b): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (b) What is the magnitude and direction of velocity of the particle at $t = 2.0 s$ ?
- Q4.21(a): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (a) At what time is the x- coordinate of the particle $16 m$? What is the y-coordinate of the particle at that time?
- Q4.21(b): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (b) What is the speed of the particle at the time ?
- Q4.22: $\hat{i}$ and $\hat{j}$ are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat{i} + \hat{j}$ , and $\hat{i} - \hat{j}$ ? What are the components of a vector $A= 2 \hat{i} + 3 \hat{j}$ along the directions of $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$ ? [You may use graphical method]
- Q4.23(a): For any arbitrary motion in space, which of the following relations are true : (a) $v_{average} = (1/2) (v (t_1) + v (t_2))$
- Q4.23(b): For any arbitrary motion in space, which of the following relations are true : (b) $v_{average} = [r(t_2) - r(t_1) ] /(t_2 - t_1)$
- Q4.23(c): For any arbitrary motion in space, which of the following relations are true : (c) $v (t) = v (0) + a t$
- Q4.23(d): For any arbitrary motion in space, which of the following relations are true : (d) $r (t) = r (0) + v (0) t + (1/2) a t^2$
- Q4.23(e): For any arbitrary motion in space, which of the following relations are true : (e) $a_{average} =[ v (t_2) - v (t_1 )] /( t_2 - t_1)$ (The ‘average’ stands for average of the quantity over the time interval $t_1$ to $t_2$)
- Q4.24(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q4.24(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q4.24(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q4.24(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q4.25: An aircraft is flying at a height of $3400 m$ above the ground. If the angle subtended at a ground observation point by the aircraft positions $10.0 s$ apart is $30^{\circ}$, what is the speed of the aircraft ?
- Q4.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q4.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q4.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q4.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q4.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q4.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q4.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q4.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q4.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q4.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q4.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q4.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q4.6(a): Establish the following vector inequalities geometrically or otherwise : (a) $|a+b| < |a| + |b|$. When does the equality sign above apply?
- Q4.6(b): Establish the following vector inequalities geometrically or otherwise : (b) $|a+b| > ||a| -|b||$. When does the equality sign above apply?
- Q4.6(c): Establish the following vector inequalities geometrically or otherwise : (c) $|a-b| < |a| + |b|$. When does the equality sign above apply?
- Q4.6(d): Establish the following vector inequalities geometrically or otherwise : (d) $|a-b| > ||a| - |b||$. When does the equality sign above apply?
- Q4.7(a): Given $a + b + c + d = 0$, which of the following statements are correct : (a) $a$, $b$, $c$, and $d$ must each be a null vector,
- Q4.7(b): Given $a + b + c + d = 0$, which of the following statements are correct : (b) The magnitude of $(a + c)$ equals the magnitude of $( b + d)$,
- Q4.7(c): Given $a + b + c + d = 0$, which of the following statements are correct : (c) The magnitude of $a$ can never be greater than the sum of the magnitudes of $b$, $c$, and $d$,
- Q4.7(d): Given $a + b + c + d = 0$, which of the following statements are correct : (d) $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of $a$ and $d$, if they are collinear ?
- Q4.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q4.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement,
- Q4.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (b) average velocity,
- Q4.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
CBSE Solutions for Class 11 Physics Motion in a Plane
Chapters in CBSE - Class 11 Physics
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per hour
Online Classes
I am excited to share my passion for English language and literature with you. With an impressive 25 years of teaching experience, I specialize in coaching for Class 10 and 11 CBSE and ICSE English, as well as spoken English. My academic journey includes an MA and a Ph.D., reflecting my dedication to the field of English studies. Inspired by my own teachers, I've been teaching English Language and Literature at various levels, both in India and abroad (Oman, Singapore, Brunei Darussalam). My commitment and excellence in teaching have been recognized through awards, including the Best Tutor Award in India and the Best Teacher Award in Brunei Darussalam. Additionally, I received a letter of commendation from the University of Cambridge Examination Board (UCLES). I offer online individual classes, where we will embark on an interactive journey through the world of English. In our sessions, I follow a structured lesson plan and use multimedia elements like PowerPoint presentations and videos to enhance the learning experience. To support your studies, I provide reference book notes. If you're ready to delve into the nuances of the English language and literature, join me in my online classes. Let's together explore the beauty of language and expression.
Top Tutor
Hari Nagar, Delhi
5 yrs of
Exp
6 students
496 hrs of classes completed
400
per hour
Online Classes
My teaching is strictly aligned with the learning objectives which a student needs to acquire over a period of time.It involves presentation of the content in an interesting way, helping the student to process it and testing him for the same.
She is very understanding and teaches statistics very well, Super sweet and very very kind teacher. Had a beautiful experience with her as a student.
