## NCERT Solutions Class 11 Science Physics Chapter 4 - Motion in a PlaneTo provide accurate answers on examinations, students must make an effort to learn each topic thoroughly. As one of the most highly rated kinematics chapters, NCERT Solutions for Class 11 Physics Chapter 4 Motion in a Plane is a crucial study resource for students in this grade. One must answer the questions in the NCERT book at the end of each chapter to do well on the Class 11 exam. Students can formulate appropriate answers using the NCERT Solutions following the CBSE Syllabus for 2022-2023's norms and marking methodology. In the previous chapter, we learned about the terms displacement, position, acceleration, and velocity. These terms are crucial for understanding how an object moves in a straight line. Using the "+" and "-" marks will take care of the direction aspect. Vectors can be used to describe how an object moves in two or three dimensions. ## Exercise:
State whether the following physical quantities are scalar or vector. - Mass
- Volume
- Speed
- Acceleration
- Density
- Number of moles
- Velocity
- Angular frequency
- Displacement
- Angular velocity
Therefore, vector quantities are acceleration, velocity, displacement, and angular velocity.
Scalar quantities include volume, mass, density, moles, and angular frequency.
From the following, pick any two scalar quantities: Force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
It is represented by,
Thus, we also know that if we apply dot product on two vector quantities, then the result is always a scalar quantity.
Magnetic Moment: An object's torque and the magnetic field are connected by a vector known as the magnetic moment. Mathematically, it can be expressed as:
The terms "work" and "current" are scalar quantities in the list provided in the question.
From the following, identify the vector quantities: Pressure, temperature, energy, time, gravitational potential, power, total path length, charge, coefficient of friction, impulse.
The given physical quantities are temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, and charge. We know that a vector quantity is obtained when a scalar quantity is multiplied by a vector quantity. Force and time, where force is a vector quantity and time is a scalar quantity, are combined to form an impulse. Impulse is, therefore, a vector quantity. Impulse is a vector quantity since it also results from adding two vector quantities.
State with reasons whether the following algebraic operations with scalar and vector physical quantities are meaningful : - Addition of any two scalars
- Adding a scalar to a vector which has the same dimensions
- Multiplying a vector by any scalar
- Multiplying any two scalars
- Adding any two vectors
- Addition of a vector component to the same vector
a.) Addition of scalar quantities is possible only if both have the same dimensions. b.) It is impossible to combine a scalar with a vector of the same dimension since a vector has both direction and magnitude while a scalar has simply magnitude. c.) It is feasible to multiply any vector by a scalar since the product of a vector multiplied by a scalar quantity is likewise a vector. d.) It is possible to multiply any two scalars, producing another scalar quantity. Think about a liquid with mass m and a temperature increase of They are both scalar quantities. The amount of heat absorbed by the liquid, which is a scalar quantity, is obtained by multiplying these numbers. e.) Only when the dimensions of the two vector quantities match can they be added. f.) We can add a component of a vector to the same vector if both of the vector quantities have the same dimension.
Read each statement below carefully and state with reasons if it is true or false: - The magnitude of a vector is always a scalar
- Each component of a vector is always a scalar
- The total path length is always equal to the magnitude of the displacement vector of a particle
- 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 the average velocity of the particle over the same interval of time
- Three vectors not lying in a plane can never add up to give a null vector
a.) True: It is accurate what is said. Displacement is a vector quantity, yet its magnitude is scalar and might be equal to the distance. b.) False: The statement is false because a vector's constituents are always other vectors. As an illustration, the motion of a bullet has both horizontal and vertical components of velocity, which are both vector numbers. c.) False: The statement is false. The overall path length might not match the displacement vector's magnitude. For instance, a circular path's circumference is the distance traveled by an object during one full rotation around the path. However, the item has zero displacements. d.) True: In cases where the velocity is negative, the average speed is positive. In this instance, average velocity exceeds average speed. It follows that the average speed is positive when the velocity is as well. The average speed and average velocity are the same in this situation. Another way to put it is that the path length is always bigger than or equal to the displacement. Consequently, the assertion is accurate. e.) True: Given a null vector, the product of the three vectors not in the same plane can be added. This is the case because they can only be added if they are on the same plane. Consequently, the assertion is accurate.
Establish the following vector inequalities geometrically or otherwise: - |a+b| < |a| + |b|
- |a+b| > ||a| −|b||
- |a−b| < |a| + |b|
- |a−b| > ||a| − |b||
When does the equality sign above apply?
a.) The parallelogram shown in the following graphic has two vectors, a and b, as its neighbouring sides. The vector a+b is shown as the parallelogram's diagonal. From parallelogram OMNP, OM = PN = |a|, OP = MN = |b|, ON = |a+b| According to the triangle's properties, each side's length is smaller than the sum of the other two sides. ∴ In ∆OMN If we substitute values in above equation,we get, |a + b| < |a| + |b| ...(i) Now if a and be act along straight line in same direction, then, |a + b|=|a| + |b| ...(ii) From Equation (i) and (ii), we get |a + b| ≤ |a| + |b| Thus, |a + b| ≤ |a| + |b|, is proved. b.) The parallelogram shown in the following graphic has two vectors, a and b, as its neighbouring sides. The vector a+b is shown as the parallelogram's diagonal. From parallelogram OMNP, OM = PN = |a|, OP = MN = |b|, ON = |a+b| According to the triangle's properties, each side's length is smaller than the sum of the other two sides. ∴ In ∆OMN ON + OM > MN |ON| > |MN - OM| If we substitute values in above equation,we get, ||a + b|| > ||a| - |b|| |a + b| > ||a| - |b|| ...(iii) Now if a and be act along straight line in same direction, then, |a + b| = ||a| - |b|| ...(iv) From Equation (v) and (iv), we get |a + b| ≥ ||a| - |b|| Thus, |a + b| ≥ ||a| - |b||, is proved. c.) The adjacent sides of the parallelogram shown in the image below is represented by the vectors a and -b, the diagonal of the parallelogram represents the vector a-b. From, parallelogram OPSR, OR = PS = |b|, OP = |a|, OS = |a-b| According to the triangle's properties, each side's length is smaller than the sum of the other two sides. ∴ In ∆OPS OS < OP + PS If we substitute values in above equation,we get, |a - b| < |a| + |b|,...(v) Now if a and be acts along the straight line in same direction, then, |a + b| = ||a| - |b|| ...(vi) From Equation (v) and (vi), we get |a - b| ≤ |a| + |b| Thus, |a - b| ≤ |a| + |b|, is proved. d.) The adjacent sides of the parallelogram shown in the image below is represented by the vectors a and -b, the diagonal of the parallelogram represents the vector a-b. From, parallelogram OPSR, OR = PS = |b|, OP = |a|, OS = |a-b| ∴ In ∆OPS OS + PS > OP, OS > OP - PS, |OS| > |OP-PS| If we substitute values in above equation,we get, ||a-b|| > ||a| - |b||, |a-b| > ||a| - |b|| ...(vii) Now if a and be acts along the straight line in same direction, then, |a - b| = ||a| - |b|| ...(viii) From Equation (vii) and (viii), we get |a - b| ≥ ||a| - |b|| Thus, |a - b| ≥ ||a| - |b||, is proved.
Given that l + m + n + o = 0, which of the given statements are true: - l, m, n and o each must be a null vector.
- The magnitude of (l + n) equals the magnitude of (m + o).
- The magnitude of l can never be greater than the sum of the magnitudes of m, n and o.
- m + n must lie in the plane of l and o if l and o are not collinear, and in the line of l and o, if they are collinear?
a.) False The statement is not correct because the summation of all the four vectors gives the result as 0 even when, b.) True Given that, l + m + n + o = 0 rearranging the above equation we get, l + n = - (m + o) ...(i) taking modulus on both sides of equation (i),we get |l + n| = | - (m + o)| (l + n) = (m + o) ...(ii) By looking at equation (ii) we can say that the magnitude of Hence, the statement given above holds true. c.) True Given that, l + m + n + o = 0 rearranging the above equation we get, l = - (m + n + o) ...(iii) taking modulus on both sides of equation (i),we get | l | = | - (m + n + o)| | l | ≤ |m| + |n| + |o| ...(iv) Now if we take a look at equation (iv) then it says that the Hence, the statement given above holds true. d.) True
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. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate?
Displacement is the distance between a particle's initial and final positions. The three girls all travel from point P to point Q. The size of the displacement is determined by the ground's diameter. Given radius is = 200 m ∴ diameter = 200 × 2 = 400 m Thus, the distance travelled by each girl is 400 m. This size is equivalent to the route that girl B skated.
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 - Net displacement
- Average velocity and
- The average speed of the cyclist
i.) Displacement is the distance between a body's original and final positions. In 20 minutes, the cyclist returns to the starting point. Therefore, there is no displacement. ii.) From i, we know that the displacement of the cyclist is zero, as a result of which the average velocity of the cyclist would also be zero. iii.)
On an open ground, a motorist follows a track that turns to his left by an angle of 600 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.
According to the given figure, the motorist travels a path which represents a hexagon with a side of 500 m. Let the starting point of the motorist be P At S,the motorist takes the third turn Therefore, magnitude of displacement is, PS = PV + VS PS = 500 + 500 PS = 1000 m Total length of the path is, PQ + QR + RS = 500 + 500 + 500 = 1500 m At Point P the motorist takes the sixth-turn, which is same as starting point. Thus, the initial and the final position of the motorist is same therefore the magnitude of displacement will become Zero. Total length of the path is, PQ + QR + RS + ST + TU + UP = 500 + 500 + 500 + 500 + 500 + 500 = 3000 m At Point R, the motorist takes the 8 The magnitude of displacement is PR, Therefore, at an angle of 30 The length of the total path is = Circumference of the hexagon + PQ + QR = 600 × 500 + 500 + 500 = 4000 m The following table displays the magnitude of the displacement and the overall length of the path according to the necessary turns:
A passenger arriving in a new town wants 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 the average speed of the taxi?
- What is the magnitude of average velocity? Are the two equal?
a.) Total distance travelled by the passenger is = 23 km Total time taken is, Therefore, b.) Total distance between the hotel and the station is = 10 km Therefore, displacement of the car is also 10 km Therefore, the two physical quantities are not equal.
Rain is falling vertically with a speed of 30 m/s. A woman rides a bicycle with a speed of 10 m/s in the north-to-south direction. What is the direction in which she should hold her umbrella?
The below figure depicts the situation given in the question. Here,
The woman must hold her umbrella toward the direction of the relative velocity (v) of the rain with respect to her in order to shield herself from the rain. As a result, the woman must hold the umbrella at a nearly 18° angle to the vertical, facing south.
A man can swim with a speed of 4 km/h in still water. How long does he take to cross a river 1 km wide if the river flows steadily at 3 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?
In a harbour, the 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?
velocity of the boat=v velocity of the wind=v The flag is moving in the direction of the northeast. It demonstrates that the wind is blowing northeastward. The flag will move in the direction of the relative wind velocity (v The Angle measured in relation to the east is equal to 45.11° - 45° = 0.11°.
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball is thrown with a speed of 40 m/s can go without hitting the ceiling of the hall?
Speed of the ball,u = 40 ms Maximum height of the ball,h = 25 m the maximum height achieved by a body projected at an angle θ.
A cricketer can throw a ball to a maximum horizontal distance of 100 m. How high above the ground can the cricketer throw the same ball?
Maximum horizontal distance,R = 100 m The angle of projection must be 45° for the cricketer to toss the ball the most horizontal distance possible. θ must be equal to 33.60° The projection velocity v horizontal range is given as, When the ball is thrown vertically upwards, it will travel the farthest. The final velocity v for such motion is zero at the maximum height H. acceleration,a = - g Following the third equation of motion we get,
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 direction and magnitude of the acceleration of the stone?
Length of the string,l = 80 cm = 0.8 m Number of revolutions = 14 Time taken is = 25 seconds The direction of centripetal acceleration is always directed along the string, toward the centre, at all points.
An aircraft executes a horizontal loop of radius 1 km with a steady speed of 900 km h
Read each statement below carefully and state, with reasons, if it is true or false: (a) The net acceleration of a particle in a circular motion is always along the radius of the circle towards the centre. (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point. (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.
(a) False Only in the case of uniform circular motion is the net acceleration of a particle moving in a circle always directed toward the centre along the circle's radius. (b) True A particle appears to move tangentially to a circular path at a certain location on the path. (c) True The direction of the acceleration vector in uniform circular motion (UCM) points in the direction of the circle's centre. But as time passes, it changes continuously. A null vector results from averaging these vectors across one cycle.
The position of a particle is given by Where t is in seconds, and the coefficients have the proper units for r to be in meters. (a) Find the 'v' and 'a' of the particle? (b) What is the magnitude and direction of the velocity of the particle at t = 2.0 s?
(a) The position of the particle is given by: (b) We have velocity vector, The presence of the negative sign denotes a velocity direction below the x-axis.
Which of the given relations is true for any arbitrary motion in space?
(a) False It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation. (b) True The arbitrary motion of the particle can be represented by this equation. (c) False The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space. (d) False The motion of the particle is arbitrary; the acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of a particle in space. (e) True The arbitrary motion of the particle can be represented by this equation.
Read each statement below carefully and state, with reasons and examples, if it is true or false. A scalar quantity is one that - is conserved in a process
- can never take negative values
- must be dimensionless
- does not vary from one point to another in space
- has the same value for observers with different orientations of axes
(a) False Energy is a scalar quantity, but in inelastic collisions, it is not preserved. b) False The temperature is a scalar quantity, however it can also have negative values. c) False A scalar quantity is the overall path length. However, it has a length dimension. d) False The value of a scalar quantity stays the same for observers with different axes. e) True For observers with various axes, the value of a scalar quantity does not change.
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°, what is the speed of the aircraft?
The aircraft is flying at a height of = 3400 m Assuming the aeroplanes are in positions A and B, the angle AOB = 30
Does a vector have a location in space? Will it fluctuate with time? Can two equivalent vectors, x and y, at various locations in space fundamentally have indistinguishable physical effects? Give cases in support of your answer.
No, Yes and No. There is no fixed point for a vector in space. This is because a vector remains invariant when it displaces in a way that prevents the change in either its direction or magnitude. Even yet, a position vector has a specific place in space. A vector changes over time. A ball moving at a certain speed, for example, experiences variations in its velocity vector over time. The physical outcome of two equivalent vectors placed at various points in space is not the same. For instance, a body will tend to rotate when two equivalent forces occur at different points on it, but their combination will not have the same turning effect.
As a vector is having both direction and magnitude, then is it necessary that if anything is having direction and magnitude, it is termed as a vector? The rotation of an object is defined by the angle of rotation about the axis and the direction of rotation of the axis. Will it be a rotation of a vector?
No and no A physical quantity that has both direction and magnitude is not always considered to be a vector. For instance, even though the current has direction and amplitude, it is a scalar quantity. The "rule of vector addition" is a fundamental need for a physical quantity to fit into the category of a vector. It follows the "rule of vector addition," which is a fundamental requirement for being a vector, but rotation of a body along an axis does not, hence it is not a vector quantity. Although in some situations, a body rotating by a little amount about an axis adheres to the law of vector addition, it is nevertheless referred to as a vector.
Can we associate a vector with - a sphere
- the length of a wire bent into a loop
- a plane area
Clarify for the same.
(i) No. We can associate the area of a sphere with the area of a vector. However, the volume of a sphere cannot be associated to a vector. (ii) No. It is impossible to connect a wire's length with a vector. (iii) Yes. A plane area can be associated to a vector.
A bullet fired at an angle of 30° 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.
The bullet is fired at an angle of = 30° The bullet hits the ground at a distance of 3 km = 3000 m The sin of an angle cannot be greater than 1. Therefore, it is impossible to hit a target that is 5 kilometres away.
A fighter plane flying horizontally at an altitude of 1.5 km with a speed of 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 ms
A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of a radius of 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?
where the symbols have their usual meaning.
(a) Let θ be the angle at which the projectile is fired with respect to x-ax θ depends on t Therefore, (b) Since, |