PART 1: Constant Angular Velocity
1. The disk rotates with a constant angular velocity → −ω of magnitude 2 rad/s. The bug continues to
rotate along a circle with a constant radius r. Since the linear velocity of an object is expressed
as →−
r ×→ −
ω , the linear velocity →
−
v also remains constant. This linear velocity remains tangential at
every point to the circle and the linear acceleration →
−
a acts towards the center of the circle along the
radius of the circle. The linear acceleration acts along the radius of the circle. This acceleration →
−
a
is generated due to the centripetal force exerted on the bug due to this rotation and this force keeps
the bug rotating along the circle.
2. The graphs are shown below:
theta vs time.png
Fig 01: Theta (angular displacement) vs time
Fig 02: Omega (angular velocity) vs time
1
, Fig 03: Alpha (angular acceleration) vs time
Screenshot of the simulator containing the graphs are shown below:
AE_01.png
Fig 04: Simulator screeshot
3. The θ vs time graph is linear. θ is the angular displacement of the bug. The relation between θ and
time is represented by th following equation:
θ = ω × t time (i)
Since the angular velocity ω is constant here, the above equation resembles the equation of a straight
line passing through the origin and the equation is y = mx where m is the slope of the line and is
constant. This is why the θ vs time graph is linear with ω as the slope of the line.
4. From equation (i), we observe that ω can be expressed as below:
θ
ω=
t
Since the unit of θ is radian (expressed as ’rad’) and unit of time is seconds (expressed as ’sec’ or
’s’), the unit of ω will be rad
s
.
The hand drawn graph is presented below.
slope_omega.png
Fig 04: Calculation of ω
2
1. The disk rotates with a constant angular velocity → −ω of magnitude 2 rad/s. The bug continues to
rotate along a circle with a constant radius r. Since the linear velocity of an object is expressed
as →−
r ×→ −
ω , the linear velocity →
−
v also remains constant. This linear velocity remains tangential at
every point to the circle and the linear acceleration →
−
a acts towards the center of the circle along the
radius of the circle. The linear acceleration acts along the radius of the circle. This acceleration →
−
a
is generated due to the centripetal force exerted on the bug due to this rotation and this force keeps
the bug rotating along the circle.
2. The graphs are shown below:
theta vs time.png
Fig 01: Theta (angular displacement) vs time
Fig 02: Omega (angular velocity) vs time
1
, Fig 03: Alpha (angular acceleration) vs time
Screenshot of the simulator containing the graphs are shown below:
AE_01.png
Fig 04: Simulator screeshot
3. The θ vs time graph is linear. θ is the angular displacement of the bug. The relation between θ and
time is represented by th following equation:
θ = ω × t time (i)
Since the angular velocity ω is constant here, the above equation resembles the equation of a straight
line passing through the origin and the equation is y = mx where m is the slope of the line and is
constant. This is why the θ vs time graph is linear with ω as the slope of the line.
4. From equation (i), we observe that ω can be expressed as below:
θ
ω=
t
Since the unit of θ is radian (expressed as ’rad’) and unit of time is seconds (expressed as ’sec’ or
’s’), the unit of ω will be rad
s
.
The hand drawn graph is presented below.
slope_omega.png
Fig 04: Calculation of ω
2
