Final Answer:
The angular acceleration of the rod needed to achieve a tangential speed of 20 m/s in 7 seconds is (b) 4.08 rad/s².
Step-by-step explanation:
To calculate the angular acceleration (( alpha )), we can use the kinematic equation:
[ v = u + a t ]
where ( v ) is the final tangential speed, ( u ) is the initial tangential speed (which is 0 since the rod starts from rest), ( a ) is the angular acceleration, and ( t ) is the time taken. Rearranging the equation to solve for ( a ):
[ a = frac{{v - u}}{{t}} ]
Substituting the given values (( v = 20 m/s ), ( u = 0 m/s ), ( t = 7 s )):
[ a = frac{{20 m/s - 0 m/s}}{{7 s}} ]
[ a = frac{{20}}{{7}} m/s² ]
Now, we know that the linear acceleration ( a ) is related to the angular acceleration ( alpha)) by the equation:
[ a = r alpha ]
where ( r ) is the radius of the circular motion. In this case, the radius is half of the rod's length (( r = frac{{text{{length of rod}}}}{2} )):
[ a = frac{{text{{length of rod}}}}{2} times alpha ]
Substituting the known values and solving for ( alpha ):
[ frac{{20}}{{7}} = frac{{ext{{length of rod}}}}{2} times alpha ]
[ alpha = frac{{frac{{20}}{{7}}}}{{rac{{text{{length of rod}}}}{2}}} ]
Given that the length of the rod is ( 20 cm = 0.2 m ):
[ alpha = frac{{frac{{20}}{{7}}}}{{0.2/2}} ]
[ alpha = 4.08 rad/s²]
Therefore, the angular acceleration is ( 4.08 rad/s²), and the correct answer is (b).