In the figure, three connected blocks are pulled to the right on a horizontal frictionless table by a force of magnitude T3 = 89.3 N. If m1 = 14.2 kg, m2 = 23.6 kg, and m3 = 32.3 kg, calculate (a) the magnitude of the system's acceleration, (b) the tension T1, and (c) the tension T2.
To solve this problem, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. We'll apply this law to each block individually.
(a) To find the magnitude of the system's acceleration, we need to find the net force acting on the system. We'll start by analyzing the forces acting on each block.
For block m1:
The only force acting on m1 is the tension T1. Since the table is frictionless, there is no force due to friction. So, we have:
T1 - m1 * g = m1 * a1,
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and a1 is the acceleration of m1.
For block m2:
The force acting on m2 is the tension T2 in the left direction and the tension T1 in the right direction. We have:
T2 - T1 = m2 * a2,
where a2 is the acceleration of m2.
For block m3:
The force acting on m3 is the tension T3 in the right direction and the tension T2 in the left direction. We have:
T3 - T2 = m3 * a3,
where a3 is the acceleration of m3.
Since all the blocks are connected and experience the same acceleration, a1 = a2 = a3 = a.
Now, we can solve the equations simultaneously to find the acceleration.
Let's rearrange the equations for each block to solve for tension:
For m1: T1 = m1 * (a1 + g),
For m2: T2 = m2 * a2 + T1,
For m3: T2 = T3 - m3 * a3.
By substituting a1 = a2 = a3 = a and rearranging, we can obtain:
T2 = T3 - m3 * a,
T1 = m1 * (a + g),
T2 = m2 * a + m1 * (a + g).
Now, we have two equations for T2. By equating them, we can solve for the acceleration:
T3 - m3 * a = m2 * a + m1 * (a + g).
Let's substitute the given values:
T3 = 89.3 N,
m1 = 14.2 kg,
m2 = 23.6 kg,
m3 = 32.3 kg,
g = 9.8 m/s^2.
By substituting these values, we get:
89.3 - 32.3 * a = 23.6 * a + 14.2 * (a + 9.8).
Now, we solve for a.
(b) To find the tension T1, we'll substitute the calculated value of acceleration (a) into the equation T1 = m1 * (a + g).
(c) To find the tension T2, we'll substitute the calculated value of acceleration (a) into the equation T2 = m2 * a + m1 * (a + g).
By solving these equations, we can find the magnitude of the system's acceleration (a), the tension T1, and the tension T2.