Physics Relating the Linear and Angular Variables?

Question

Wheel A of radius rA = 10 cm is coupled by a belt B to wheel C of radius rC= 25 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s^2. Find the time needed for wheel C to reach an angular speed of 100 rev/min, assuming the belt does not slip. (Hint: If the best does not slip, the linear speeds at the two rims must be equal.)

Please help me with this problem I really want to learn how to solve it because I have an exam next week and I don't understand this! Thanks so much!

Answer 1

Ok. From the hint start by figuring out the ratio of wheel speeds
Circumference of A =2*pi*10
Circumference of C = 2*pi*25
so C/A=2.5 ie A must go around 2.5 times to make C go around 1 time.
There are 2*pi radians to 1 revolution so 100 rev/min =200*pi rad/min. A must go 2.5 times as fast as this or 2.5*200*pi=500 rad/min
Accelleration is the derivative of velocity which means that we must integrate the angular acceleration to obtain the speed at any given point of time.
Integrating 1.6rad/s^2 with respect to time and setting velocity equal to zero at time equals zero we obtain angular velocity= 1.6*time elapsed (rad/sec). Note that this speed is in "per seconds" and we need it to be in "per minutes" so we must multiply by 60 for the units conversion.
This gives us the angular speed in rad/min for any chosen passage of time. We shall set this equal to the goal speed of 500*pi rad/min as mentioned above. So
1.6*time elapsed*60 =500*pi
Solving for time elapsed we obtain a time of 1.63 minutes.

Answer 2

The linear velocity is (2 pi r x rotation velocity) so if C has speed of100 rev/m , A has speed 25/10 x 100 rev/m. =1.6 rad/sec^2 x time
Solve for time