The figure here shows four penguins that are being playfully pulled along very slippery (frictionless) ice by a curator. The masses of three penguins and the tension in two of the cords are m1 = 12 kg, m3 = 16 kg, m4 = 21 kg, T2 = 101 N, and T4 = 202 N. Find the penguin mass m2 that is not given.
without a figure, it is not possible to decipher this.
To find the penguin mass m2, we will use Newton's second law, which states that the acceleration of an object is directly proportional to the force applied and inversely proportional to the mass of the object (F = ma).
First, let's consider the forces acting on each penguin:
- For penguin m2, it experiences two forces: tension T2 acting to the right and tension T3 acting to the left.
- For penguin m1, it experiences two forces: tension T1 acting to the right and tension T2 acting to the left.
- For penguin m3, it experiences two forces: tension T3 acting to the right and tension T4 acting to the left.
- For penguin m4, it experiences one force: tension T4 acting to the right.
Since the ice is frictionless, there is no horizontal force other than the tensions. Therefore, we can set up the following equations for the net forces on each penguin in the horizontal direction:
For penguin m2:
T2 - T3 = m2a2
For penguin m1:
T1 - T2 = m1a1
For penguin m3:
T3 - T4 = m3a3
For penguin m4:
T4 = m4a4
Now, we have two tensions and three accelerations. To find the unknown mass m2, we need to eliminate the unknown tensions.
Since we know the value of T2 (101 N) and T4 (202 N), we can substitute them into their respective equations:
From T2 - T3 = m2a2, we have 101 N - T3 = m2a2.
From T4 = m4a4, we have 202 N = m4a4.
Now, we can substitute T4 in the equation for penguin m3: T3 - T4 = m3a3.
Substituting the values, we have T3 - 202 N = m3a3.
We can rearrange this equation to solve for T3: T3 = m3a3 + 202 N.
Now, we can substitute this value for T3 in the equation for penguin m2:
101 N - (m3a3 + 202 N) = m2a2.
Simplifying this equation, we get:
-101 N - m3a3 = m2a2.
Since we have values for m1, m3, a3, T2, and T4, we can substitute them into the equation, solving for m2:
-101 N - (16 kg)(a3) = (12 kg)(a2).
Simplifying further, we have:
-101 N - 16 kg(a3) = 12 kg(a2).
Now, we need to find the value of a2.
To find the acceleration a2, we can consider the entire system of penguins as a single object. Since all the forces act in the horizontal direction, we can sum up the masses and tensions to find the acceleration of the system:
Sum of forces = Total mass × Acceleration.
The sum of the forces is T1 - T4, and the total mass is m1 + m2 + m3 + m4.
So, we have:
T1 - T4 = (m1 + m2 + m3 + m4) × a.
Substituting the values we have for T1 and T4, we get:
T1 - 202 N = (12 kg + m2 + 16 kg + 21 kg) × a.
Simplifying further, we have:
T1 - 202 N = (m2 + 49 kg) × a.
Now, substituting the value of a from this equation into the previous equation for m2, we get:
-101 N - 16 kg(a3) = 12 kg(a2).
Replacing a2 with (T1 - 202 N) / (m2 + 49 kg), we have:
-101 N - 16 kg(a3) = 12 kg((T1 - 202 N) / (m2 + 49 kg)).
Now, we can solve this equation for m2.