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Chapter 2
Units and Measurement
1.Name the fundamental(base) quantities and units according to SI system.
2.Define angle
3.Define solid angle
4.Write the dimensional formulae of following derived quantities.
Area - L
2
Work or energy - ML
2
T
− 2
Volume - L
3
Power - ML
2
T
− 3
Density - ML
− 3
Pressure - ML
− 1
T
− 2
Velocity- LT
− 1
Stress - ML
− 1
T
− 2
Acceleration - LT
− 2
Modulus of elasticity- ML
− 1
T
− 2
Momentum - MLT
− 1
Force - MLT
− 2
5.Write two physical quantities having no unit and dimension
Relative density, strain
Seema Elizabeth, MARM Govt HSS Santhipuram, Thrissur
6.Write two physical quantities that have unit but no dimension
Plane angle, solid angle, angular displacement 7.
9.Name and state the principle used to check the correctness of an equation. Principle
of homogeneity of dimensions.
For an equation to be correct the dimensions of each terms on both sides of the
equation must be the same
Or
The magnitudes of physical quantities may be added or subtracted only if they
have the same dimensions
- Using the method of dimension check whether the equation is dimensionally
correct or not
Since the dimensions of all terms are not same the equation is not correct
𝒂
16.The Van der waals equation of 'n' moles of a real gas is (P+𝑽𝟐)(V−b)=nRT. Where P is
the pressure, V is the volume, T is absolute temperature, R is molar gas constant and a, b,
c are Van der waal constants. Find the dimensional formula for a and b.
𝐚
(P+𝐕
𝟐
)(V−b)=nRT.
By principle of homegeneity, the quantities with same dimensions can be added or
subtracted.
a
[P] =[V 2 ]
[a] =[PV
2
]
=ML
− 1
T
− 2
x L
6
[a] = ML
5
T
− 2
[b] = [V]
[b] =L
3
17.Derive the equation for kinetic energy E of a body of mass m moving
with velocity v
18.Suppose that the period of oscillations of a simple pendulum depends on its mass of
the bob(m),length(l) and acceleration due to gravity(g).Derive the expression for its
time period using the method of dimensions.
19.Write any two limitations of dimensional analysis.
20.Find the number of significant figures in following numbers
4.700 × 10
2
4.700 × 10
− 3
21.If mass of an object is measured to be, 4.237 g (four significant figures) and its
volume is measured to be 2.51cm3(3 significant figures), then find its density in
appropriate significant figures.
𝑔
Density == = 1.
T=k √
𝑙
𝑔
𝑚𝑎𝑠𝑠 4.
𝑣𝑜𝑙𝑢𝑚𝑒 2.
6.. The position - time graph of an object in uniform motion
is------------- Ans: A straight line inclined to the time
axis
- The slope of position-time graph gives -------------
Ans: Velocity
- The velocity - time graph of an object in uniform motion is-------------
A straight line parallel to the time axis
- The area under velocity - time graph gives --------------
Displacement
- The slope of velocity-time graph gives ---------------
Acceleration
- Define average accelaration
The average acceleration over a time interval is defined as the ratio of change in
velocity to the time interval.
𝐯𝟐−𝐯𝟏
𝚫𝐯
𝐭𝟐−𝐭𝟏 𝚫𝐭
12.Draw the position- time graph of an object moving with
- Draw the velocity- time graph of an object moving with
(a) uniform positive acceleration (b) uniform negative acceleration
14.Draw the velocity- time graph of a stone thown vertiaccly upwrds and comes back.
15.Draw the speed- time graph of a stone thown vertiaccly upwrds and comes back.
- Draw the velocity-time graph of a freely falling body.( A stone vertically falling
downwards)
- An object is under freefall. Draw its (a) Acceleration - time graph (b)
Velocity- time graph
(c) Displacement-time graph
21.Velocity – time graph of a body is given below
a) Which portion of the graph represents uniform retardation?
(i) OA (ii)AB (iii) BC (iv) OC
b) Find the displacement in time 2s to 7s.
c) A stone is dropped from a height h. Arrive at an expression for the time
taken to reach the ground.
a)BC b)
Displacement = area of rectangle
. = 6 x 5 =
30m
c) s = ut +½ at
2
2h
|𝐀|
𝐀
Motion in a Plane
- Differentiate scalar and vector quantities
A scalar quantity has only magnitude and no direction.
Eg. distance , speed, mass , temperature, time ,work ,power, energy, pressure,
frequency, angular frequency etc.
A vector quantity has both magnitude and direction and obeys the triangle law
of addition or the parallelogram law of addition.
Eg. displacement, velocity, acceleration , momentum, force,
angular velocity, torque, angular momentum etc.
2.When two vectors are said to be equal?
Two vectors A and B are said to be equal if, and only if, they have the same
magnitude and the same direction.
3.What do you mean by null vectors or zero vector?
A Null vector or Zero vector is a vector having zero magnitude and is represented by O
or Ō. The result of adding two equal and opposite vectors will be a Zero vector Eg:
When a body returns to its initial position its displacement will be a zero vector.
4.What are unit vectors?
A unit vector is a vector of unit magnitude and points in a particular direction. It
has no dimension and unit. It is used to specify a direction only.
5.The position vector of a particle P located in an x-y plane is shown in figure.
a)Redraw the figure by showing the rectangular components.
b)Write the position vector in terms of rectangular components.
a)
b) 𝑟̅ = 𝑟̅ cos 𝜃𝑖̂ + 𝑟̅ sin 𝜃𝑗
6.State triangle law of vector addition.
If two vectors are represented in magnitude and direction by the two sides of a triangle
taken in order ,then their resultant is given by the third side of the triangle taken in
reverse order.
7.State parallelogram law of vector addition
If two vectors are represented in magnitude and direction by the adjacent sides of
a parallelogram ,then their resultant is given by the diagonal of the parallelogram.
8.Two vectors A and B
are given
Redraw the figure and
show the vector sum
using
9.Write the equation to find the magnitude of resultant of two vectors A and B
R
10.Derive the expression for magnitude of resultant of two vectors by analytical method.
SNP , cos θ = PN / PS sin θ = SN /PS
cos θ = PN / B sin θ = SN / B
PN = B cos θ SN = B sin θ
From the geometry of the figure,
OS
OS (OP + PN)
OS = (A + B cos θ )
R +2AB cos θ + θ +B
below.
parallelogram method.
Write the expression for direction of
resultant vector.
s = ut +½ at
2
s=0, u = u sin θ , a =-g , t = T
0 = u sin θ T - ½ gT
2
½ gT 2 =u sin θ T
2 u sin θ
T=
g
Horizontal range of a projectile (R)
Horizontal range = Horizontal component of velocity x Time of flight
2 u sin θ
R = u cos θ x g
u
2
x 2 sinθ cos θ
R = g
u
2
sin 2θ
R =
g
Maximum height of a projectile (H)
Consider the motion in vertical direction to the highest point
v
2
2
= 2as
u = u sin θ, v = 0 , a = - g , s = H
0 - u
2
sin
2
θ = - 2 g H
u
2
sin
2
θ
H =
2g
18.What is the angle of projection for maximum horizontal range
𝟎
- What is the maximum value of horizontal range
Range is maximum when θ=
0
u
2
sin 90
R
R
max
g
- Find the angle of projection for which the range will be same as that in case of
θ=𝟑𝟎
𝟎
for a given velocity of projection.
For a given velocity of projection range will be same
for angles 𝜽 and ( 90-𝜽 )
Here θ=
0
0
The range will be same for 30
0
and 60
0
,for a given velocity of projection.
21.A cricket ball is thrown at a speed of 28 m s
in a direction 30° above the
horizontal. Calculate (a) the maximum height, (b) the time taken by the ball to return
to the same level, and (c) the distance from the thrower to the point where the ball
returns to the same level.
(a) H = u
2
sin
2
θ
2g
H = 28
2
sin
2
2 x 9.
H = 10 m
(b) T = 2 u sin θ
g
T = 2x 28 sin
g
u
2
angle = Δ
θ =
Δ r = r Δ θ
r
Linear
velocity v =
r v =
𝑡
But ω =
𝑡
v = r ω
29.Define angular acceleration The rate of change of angular
velocity is called angular acceleration. d ω
α = dt
d θ
ω = d𝑡
d d θ
α = ( )
dt 𝑑𝑡
𝑑
2
𝜃
2
30.Define Centripetal acceleration
A body in uniform circular motion experiences an acceleration , which is directed
towards the centre along its radius .This is s called centripetal acceleration.
Δ
Δ θ
r
𝑡
arc
radius
Δ r
31.Derive the expression for centripetal acceleration.
= v r v Δ r
Δ v
v a = x r
r
𝐯𝟐
a =
𝐫
If R is the radius of circular path, then centripetal acceleration.
v 2
a c
R
a c
= ω
2
R
ac = v ω
32.An insect trapped in a circular groove of radius 12 cm moves along the groove
steadily and completes 7 revolutions in 100 s.
(a) What is the angular speed, and the linear speed of the motion?
(b) Is the acceleration vector a constant vector? What is its magnitude?
Period, T= s
(a) The angular speed ω is given by
2π 2π
ω = = = =0.44 rad/s
𝑇
The linear speed v is :
Δ v Δ r
Δ Δ
r
v v r
t r t
