A disk rotates at constant angular acceleration, from angular positionθ1=10.0 rad to angular positionθ2=70.0rad in 6.00s. Its angular velocity atθ2 is15.0 rad/s. (a) What was its angular velocity at θ1? (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph θversus time tand angular speedω versust for the disk, from the beginning of the motion (lett=0 then).

Short Answer

Expert verified

a) Angular velocity atθ1 isω1=5.00 rad/s

b) Angular acceleration isα=1.67 rad/s2

c) Angular position when disk was at rest isθ0=2.50 rad

d) Graphs of θVs t andω Vs t are drawn below

Step by step solution

01

Given

i) Angular positionθ1=10.0rad

ii) Angular positionθ2=70.0 rad

iii) Time ist=6.00 s

iv) Angular velocity at θ2is15.0 rad/s

02

Understanding the concept

Use the concept of angular kinematics. Use the angular displacement equation to find the angular velocity at the given theta. Using the angular kinematic equation, which relates initial and final angular velocities and acceleration, find angular acceleration. Draw the graph of angular displacement versus t and angular velocity versus t.

Formulae:

θθ0=12(ω0+ω)t................(i)

ω=ω0+αt...........................(ii)

ω2=ω02+2α(θθ0).............(iii)

03

(a) Calculate the angular velocity at θ1 

Angular velocity at :θ1

We can use the equation (i) as follows:

θ2θ1=12(ω1+ω2)t7010=12(ω1+15.0)(6.00)60×26.0015.0=ω1ω1=5.00rad/s

Hence, the angular velocity is.5.00rad/s

04

(b) Calculate the angular acceleration

Angular acceleration:

We use the equation (ii) to find angular acceleration as follows:

15.0=5.00+α(6.00)α=15.05.006.00=1.66 rads2=1.67 rad/s2

Hence, the angular acceleration is1.67 rad/s2.

05

(c) Calculate the angular acceleration

Angular position when disk was at rest:

We have the equation (iii) as follows:

ω12=ω02+2α(θ1θ0)5.02=02+2(1.67)(10.0θ0)θ0=252×1.67+10.0=2.50 rad

Hence, the angular acceleration is.2.50 rad

06

(d) Graph θversus time tand angular speedω versus t for the disk

Graph ofθ Vs t andω Vs t:

We can draw the given graphs as follows:

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