In order to find the acceleration of the center of mass of the system, which would be the system including 25kg block, string and 47kg block, you first have to find the acceleration of individual blocks. For this question, let's set the counterclockwise direction as positive. Also, I will use m1 for the 25kg block and m2 for the 47kg block.
Newton's second law for m1:
∑F1 = m1a
T1 - m1g = m1a
T1 = m1g + m1a = (25)(9.8) + 25a
Newton's second law for m2:
*** Remember, as we set the counterclockwise direction as positive, the block would have a positive acceleration despite accelerating downwards.
∑F2 = m2a
m2g - T2 = m2a
T2 = m2g - m2a = (47)(9.8) - 47a
Newton's second law (rotationally) for the disk:
∑τ= I*α
rT2 - rT1 = 1/2(Mr^2)*(a/r)
T2 - T1 = 1/2(69)(a)
Now, we can substitute values to solve for a:
(47)(9.8) - 47a - [(25)(9.8) + 25a] = 1/2(69)(a)
a = 2.024413146 m/s^2
Now, using this acceleration of the individual blocks, we can solve for the acceleration of the center of mass of the system (acm) using the equation m1a + m2a = (m1+m2)acm. Now, as m1 is accelerating upwards, it would have a positive acceleration and as m2 is accelerating downwards, it would have a negative acceleration.
Hi,
Thank you for your question.
In order to find the acceleration of the center of mass of the system, which would be the system including 25kg block, string and 47kg block, you first have to find the acceleration of individual blocks. For this question, let's set the counterclockwise direction as positive. Also, I will use m1 for the 25kg block and m2 for the 47kg block.
Newton's second law for m1:
∑F1 = m1a
T1 - m1g = m1a
T1 = m1g + m1a = (25)(9.8) + 25a
Newton's second law for m2:
*** Remember, as we set the counterclockwise direction as positive, the block would have a positive acceleration despite accelerating downwards.
∑F2 = m2a
m2g - T2 = m2a
T2 = m2g - m2a = (47)(9.8) - 47a
Newton's second law (rotationally) for the disk:
∑τ= I*α
rT2 - rT1 = 1/2(Mr^2)*(a/r)
T2 - T1 = 1/2(69)(a)
Now, we can substitute values to solve for a:
(47)(9.8) - 47a - [(25)(9.8) + 25a] = 1/2(69)(a)
a = 2.024413146 m/s^2
Now, using this acceleration of the individual blocks, we can solve for the acceleration of the center of mass of the system (acm) using the equation m1a + m2a = (m1+m2)acm. Now, as m1 is accelerating upwards, it would have a positive acceleration and as m2 is accelerating downwards, it would have a negative acceleration.
(25)(2.024413146) - (47)(2.024413146) = (25+47)acm
acm = - 0.6185706835 m/s^2
Hence, the acceleration of the center of mass of the system would be - 0.6185706835 m/s^2.
Does this answer your question?