Chapter No.2 Short QuestionsF.Sc 1st Year Vectors and EquilibriumQ.No.2.1: - Define the terms. (i) Unit vector (ii) Position Vector (iii) Components of a vector.Ans: - (I) Unit Vector: - A vector whose magnitude is unity is called unit vector. A unit vector in the direction of vector A is obtained by dividing the vector A by its magnitude A thus,

(ii) Position Vector: - The position vector is a vector that describes the location of a particle with respect to the origin. The position vector of a r of a point P (a, b, c) in space is

(iii) Component of a Vector: - A component of a vector is its effective value in a given direction. A vector can be consid-ered as the resultant of its component vectors along the specified directions. Q.No.2.2: - The vector sum of three vectors gives a zero resultant. What can be the orientation of the vectors?Ans: - When more than two vectors combine using head to tail rule in such a way that they form a closed shape then the resultant of these three vectors will be zero. We can easily make a triangle by adding three vectors. So their sum will be zero. As shown in Fig;

Thus,

Q.No.2.3: - Vector A lies in the xy-plane. For what orientation will both of its rectangular comp- onent be negative? For what orientation will its components have opposite signs?Ans: - In 1st quadrant x and y are both +ve. In 2nd x is –ve and y is +ve. In 3rd both x and y are –ve. In 4th quadrant x is +ve and y is –ve. By using this consideration we can say that when vect-

or A lies in 1st quadrant then both of its rectangular components will have same +ve signs when the vector A lies in 2nd quadrant then x component is –ve but y will be +ve. When A lies is 3rd

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quadrant then both components will have same negative signs and in the 4th quadrant the x and y will have opposite signs means x is +ve and y is –ve as shown in Fig,

Q.No.2.4: - If one of the rectangular components of a vector is not zero, can its magnitude be zero? Explain.Ans: - No. its magnitude can never be zero.Example: - We know that the magnitude of a vector F can be given as;

This shows that if one of the components is zero then the vector itself will not be zero. ( Because there is a sum

of rectangular components if there was a product then the vector will be zero but taking one of its components zero.)Q.No.2.6: - Can the magnitude of a vector have a negative value?Ans: - No, the magnitude of a vector cannot be a negative value it always has a positive value.

The magnitude of a vector is a scalar. As we know that a vector in 2-D have x and y component generally but its has a square root which have ± value means the vector may have a negative or a positive value but negative magnitude have no sense so we always take the positive value.(Conceptual example: - For example height is a scalar quantity if we build a house then its height is always in upward direction means + direction if we say that we build a house in downward direction then what does it mean it seems very funny it has no proper sense.)Q.No.2.7: - If A+B=0, what can you say about the component of the two vectors?Ans: - As given that the given two vectors are adding to each other so there component will also add. From vector algebra we know that the sum of vectors will be zero only if half vector or

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their components are positive and the other half have same magnitude but opposite in direction to the other half components. Mathematically;

As all the components of both the vectors are equal but opposite so they will cancel each other and the result will be zero.Q.No.2.8: - Under what circumstances would a vector have components that are equal in magnitude?Ans: - In case of rectangular components, the magnitude of the two components will be equal if the vector makes an angle of 45° with +x-axis. As shown in the fig,Mathematically;

Q.No.2.9: - Is it possible to add a vector quantity to a scalar quantity? Explain.Ans: - No, it is not possible to add vector to a scalar quantity. We can add vectors only like quan-tities. Different physical quantities can’t be added. On the other hand we can say that vectors can be added by head to tail rule because vectors have direction while scalar do not have any direction. So we can’t add vector to a scalar.Q.No.2.10: - Can you add zero to a null vector?Ans: - It’s natural to say yes at first because both have same magnitude i.e.; 0. But when we thought deeply then we arrive at conclusion that null vector is a vector it’s mean it must have direction no matter in which direction it is. So, we said that it’s impossible to add zero to a null vector because both are different physical quantities.Q.No.2.11: - Two vectors have unequal magnitudes. Can their sum be zero? Explain.Ans: - No, the sum of two unequal vectors cannot be zero. Two vectors of unequal magnitudes can never be combined to give zero resultant whatever the orientation may be. However, the sum of two vectors can be zero if both of them are equal and in opposite directions.Q.No.2.12: - Show that the sum and difference of two perpendicular vectors of equal lengths are also perpendicular and of the same length.

Ans: - Let us suppose there are two vectors A and B which are equal in magnitude and perpend-icular to each other as shown in the figure. According to head to tail rule,

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Hence we can say that R is perpendicular to Ro. Therefore, both sum and difference are perpend-icular to one another. Now by using the formula for magnitude of rectangular components, we have

Q.No.2.13: - How would the two vectors of the same magnitude have to be oriented, if they were to be combined to give a resultant equal to a vector of the same magnitude?

Ans: - If the angle between two vectors of same magnitude is 120º, the magnitude of their resultant vector is also the same.This can be proved as follows; Let F

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Q.No.2.14: - The two vectors to be combined have magnitudes 60N and 35N. Pick the correct answer from those given below and tell why it is the only one of the three that is correct.(i) 100N (ii) 70N (iii) 20NAns: - When we combine to or more vectors then we can do that only by vector addition using head to tail rule. So if we add these to 60N and 35N forces then we get 95N resultant force. The resultant can not exceed 95N. If we subtract these forces then the resultant we become 25N and similarly the resultant will not be less than 25N so the answer is between these to resultants i.e.; 70N. The option (ii) is correct.Q.No.2.15: - Suppose the sides of a closed polygon represent vector arranged head to tail. What is the sum of these vectors?Ans: - We know that the resultant of a number of vectors which make a closed path is equal to zero. Let the vectors A,B,C,D and E are represented by the sides of a closed polygon, then their sum will be zero because the head of the last vector i.e.; E coincides with the tail of the first vector i.e.; A. Hence A+B+C+D+E=0(Note that on R.H.S, zero (0) is also a vector not a scalar because vector addition always results in a vector.)

Q.No.2.16: - Identify the correct answer:(i) Two ships X and Y are traveling in different directions at equal speeds. The actual direction of motion of X is due north but to an observer on Y, the apparent direction of motion of X is north-east. The actual direction of motion of Y as observer from the ship will be (A) East (B) West (C) South-east (D) South-west.(ii) A horizontal force F is applied to a small object P of mass m at rest on a smooth plane inclined at an angle θ to the horizontal as shown in Fig. The magnitude of the resultant force acting up and along the surface of the plane, on the object is

Ans: - The correct answer is (B) i.e; west.Proof: - The problem will be solved with the help of relative velocity. The relative velocity of

ship X with respect to ship Y is . In diagram OA indicates the velocity vector of ship X

in north direction. The relative velocity vector OB i.e., of ship X with respect to ship Y

makes an angle of 45° with east direction as shown in the figure. Now join A with B.

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From the figure, we have

Now AB is towards east, so velocity would be in opposite direction (being negative of AB).

It will be towards west. Hence the correct answer is (B) west.

(ii) The correct answer is (a) Proof: - In the figure, the weight (w=mg) of the obj-ect and force F is resolved into rectangular compon-ents. The weight w is resolved into two rectangular components.

(i) mgsinθ (along the plane)(ii) mgcosθ (perpendicular to the plane)

From figure mgcosθ = RNow force F is resolved into two rectangular components.

(i) Fcosθ acting up along the surface(ii) Fsinθ Perpendicular to the plane

From figure, the reaction R is balanced by Therefore, the net force is zero. Hence the magnitude of the resultant force acting up and along the surface of the plane will be given by Which is the true answer.

Q.No.2.17: - If all the components of the vectors, were reversed,

how would this alter

Ans: - If all the components of the vectors are reversed i.e., then

The order in cross multiplication matters only when the components of one of the two vectors were reversed. But when both the vectors have reversed components then the result remain

unaltered. That’s why .

Q.No.2.18: - Name the three different conditions that could make

Ans: - can be zero only if,

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Hence

Q.No.2.19: - Identify true or false statements and explain the reason.(a) A body in equilibrium implies that it is not moving nor rotating.(b) If coplanar forces acting on a body form a closed polygon, and then the body is said to be in equilibrium.

Ans: - (a) This is false statement since a body may be in equilibrium if it is moving with uniform

velocity.(b) This is a correct statement since a closed polygon represents zero vector sum of forces

and this is the 1st condition of equilibrium.Q.No.2.20: - A picture is suspended from a wall by two strings. Show by diagram the configuration of the strings for which the tension in the strings will be minimum.Ans: - The tension in the suspension strings will be minimum for the configuration shown in the figure. In this case, the tension in each string will be half that of weight. 2T = W or T = W/2

Q.No. 2.21: - Can a body rotate about its center of gravity under the action of its weight?

Ans: - We know that torque is equal to where r is the moment arm which is the distance from the origin to the point of application of force when we want to find the torque or rotating effect of a body about the center of gravity under the action of its own weight (in this case the force applied is weight). The point of application of force (weight) is at the center of gravity so moment arm is zero (by definition) so . This means no rotaion.

Prepared by Muzammal Hussain. Email: - muzammalsafdar@gmail.comVisit http://www.fscphysicsofpk.blogspot.com/ for online access

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