Scalars and vectors

Distinguish between a scalar and a vector quantity

A quantity which can be described fully by giving its magnitude and unit is known as a scalar quantity. They can be added algebraically.

A vector quantity has magnitude, unit and direction. They may not be added algebraically.

A mass of weight 120N is hung from two strings as shown in Figure below. Determine, by scale drawing or by calculation, the tension in:

i. RA,

ii. RB.

Since they balance each other out, the horizontal components of the rope's tension add up to zero or equal. Hence,

`T1x=T2x =>T1xxcos(50^o)=T2xxcos(40^o)`

`=>T1=(cos(40^o))/(cos(50^o))xxT2=1.192xxT2`

The sum of vertical components of the tension is equal to the weight of the block. That means that

`T1y+T2y=120N`

`=> T1xxsin(50^o)+T2xxsin(40^o)=120N`

`1.192xxT2xxsin(50^o)+T2xxsin(40^o)=120N`

`=>T2=RB=77.125~~77N`

`=>T1=RA=1.192xx77.125=91.933~~92N`

A fielder in a cricket match throws the ball to the wicket-keeper. At one moment of time, the ball has a horizontal velocity of 16 `m.s^(-1)` and a velocity in the vertically upward direction of 8.9 `m.s^(-1)`.

a. Determine, for the ball:

    i. its resultant speed

    ii. the direction in which it is travelling relative to the horizontal

b. During the flight of the ball to the wicket-keeper, the horizontal velocity remains unchanged. The speed of the ball at the moment when the wicket-keeper catches it is 19 `m.s^(-1)`. Calculate, for the ball just as it is caught:

    i. its vertical speed,

    ii. the angle that the path of the ball makes with the horizontal.

c. Suggest with a reason whether the ball, at the moment it is caught, is rising or falling.

a. i. Resolve the velocities into horizontal and vertical components as figure above

Given: 

Horizontal velocity, `v_x=16 m.s^(-1)`

Vertical velocity, `v_y=8.9m.s^(-1)`

To calculate the resultant speed of the ball, the Pythagorean theorem is applied then the resultant speed is given by:

`v=sqrt(v""_x^2+v""_y^2) =sqrt(16^2+8.9^2)=18.31 m.s^(-1)`

ii. To determine the direction of the ball relative to the horizontal, the angle θ that the path of the ball makes with the horizontal is given by:

`theta=arctan(v_y/v_x)=arctan(8.9/16)=29^o`

Thus, since the vertical speed is positive, the ball is rising at the moment it is caught.

b. i. To calculate the vertical speed of the ball, the Pythagorean theorem is applied, and then the vertical speed is given by:

`v_y=sqrt(v^2-v""_x^2)=sqrt(19^2-16^2)=10.25m.s^(-1)`

ii. To determine the direction of the ball relative to the horizontal, the angle θ that the path of the ball makes with the horizontal is given by:

`theta=arctan(v_y/v_x)=arctan(10.25/16)=32.64^o`

c. Thus, since the vertical speed is positive, the ball is rising at the moment it is caught.

The velocity vector diagram for an aircraft heading due north is shown to scale in the figure below. There is a wind blowing from the north-west. The speed of the wind is 36 `m.s^(-1)` and the speed of the aircraft is 250 `m.s^(-1)`.

i. Make a copy of the figure below. Draw an arrow to show the direction of the resultant velocity of the aircraft.

ii. Determine the magnitude of the resultant velocity of the aircraft.

i. 

ii. The cosine formula is applied,

`v^2=v""_"wind"^2+v""_"aircraft"^2-2.v_"wind".v_"aircraft".costheta`

`=> v=sqrt(36^2+250^2-2.36.250.cos(45^o)) =226m.s^(-1)`

Which of the following is a vector?

A. electric charge 

B. momentum 

C. power 

D. work

The answer is B.

Scalars are quantities that have magnitude and unit only; they are independent of direction. Vectors have all three of them. Momentum is the product of the mass and velocity of an object p = mv. Momentum is a vector quantity since it has a direction as well as a magnitude. The rest of quantities in option pressure work and electric charge have magnitude but not direction.

A girl walks at a speed of 1.5 `m.s^(-1)` for 1.0 minutes in a direction of 35° north of east. How far east does she walk?

A. 52m 

B. 63m 

C. 74m 

D. 90m

The answer is C.

Convert 1.0 minute to seconds: 1.0 minute = 60 seconds. 

Calculate the horizontal component of the girl's velocity:`v_x=vxxcos(theta)` where, `v=1.5 m.s^(-1)` and `theta=35^(o)`.

`v_x=1.5xxcos(35^o)=1.2287 m.s^(-1)`

Calculate the distance traveled east:

`d_"east"= v_x xx t` where `t=60s`

Thus,

`d_"east"=1.2287xx60=73.722m~~74m`

Two physical quantities P and Q are added. Sum of P and Q is R, as shown. Which quantity could be represented by P and by Q?

A. Kinetic

B. Power

C. Speed

D. Velocity

The answer is D.

Since P and Q are added to give R, we can represent this situation as: `vec P + vec Q =vec R`. 

From the given options, kinetic energy, speed and power (rate of doing work) are scalar quantities whereas velocity is vector quantity. When adding two vectors, the result can be another vector. Therefore, P and Q could be represented by vector quantities.

Which list contains only scalar quantities? 

A. area, length, displacement 

B. kinetic energy, speed, power 

C. potential energy, momentum, time 

D. velocity, distance, temperature

The answer is B.

Vector Quantity: Displacement, Momentum, Velocity 

Scalar Quantity: Area, Length, Kinetic energy, Speed, Power, Potential energy, Time, Distance, Temperature.

The diagram shows a displacement vector.

What is the vertical component of this displacement vector?

A. 3.0 km

B. 4.0 km

C. 5.0 km

D. 6.6 km

The answer is B.

`d_v=5.0xxsin(53^o)=4.0 km`

Which pair of quantities contains both a scalar and a vector? 

A. Impulse and angular momentum 

B. Work and frequency 

C. Electromagnetic force and force 

D. Current density and current

The answer is C.

Option A: Impulse and angular momentum. Impulse is defined as the change in momentum of an object, which is equal to the product of force and the time interval. Angular momentum is defined as the product of mass linear velocity and radius and it is not a scalar quantity nor a vector quantity. 

Option B: Work and frequency, The work is defined as the product of force and displacement and work is a scalar quantity as the dot product of force and displacement is always a scalar quantity. Frequency is defined as the number of oscillations made per second. Frequency is a scalar quantity. 

Option C: Electromagnetic force and force. Electromagnetic force is defined as the energy per unit charge and is a scalar quantity. Electromagnetic force is a scalar quantity. Force is defined as the product of mass and acceleration and force is a vector quantity. This is the pair in which one physical quantity is scalar and other is vector. The correct answer is electromagnetic force and force. 

Option D: Current density and current. Current is defined as a moving stream of charged particles ions and electrons from one point to another point. Current is a scalar quantity. Current density is defined as the amount of charge per unit time that flows through a unit area of cross section. The current density is vector quantity.