Three blocks of masses M1=26kg,M2=38kg and M3=41kg are connected by 2 strings over 2 pulleys.Friction is negligible.
1. Determine the magnitude of acceleration
and 2. the magnitude of the tension in each of two strings
.
1. The magnitude of acceleration is given by the equation a = (M1 + M2 - M3)g/ (M1 + M2).
Therefore, a = (26 + 38 - 41)g/(26 + 38) = 0.8g.
2. The magnitude of the tension in each of the two strings is given by the equation T = (M1 + M2)a.
Therefore, T = (26 + 38)0.8g = 64g.
To determine the magnitude of acceleration and the magnitude of tension in each of the two strings, we need to apply Newton's laws of motion.
1. Magnitude of Acceleration:
Considering the three blocks together as a system, the net force acting on the system will determine the acceleration. The net force can be calculated by summing up all the individual forces acting on the system.
a. First, let's identify the forces acting on each block:
- For M1 (26 kg):
- The tension force (T1) acting to the right.
- The force due to the weight (W1) acting vertically downward.
- For M2 (38 kg):
- The tension force (T2) acting to the left.
- The force due to the weight (W2) acting vertically downward.
- For M3 (41 kg):
- The force due to the weight (W3) acting vertically downward.
b. Now, let's calculate the net force:
- For M1:
- T1 - W1 = M1 * a (equation 1)
- For M2:
- T2 - W2 = M2 * a (equation 2)
- For M3:
- W3 = M3 * a (equation 3)
c. Combining equations 1, 2, and 3:
- T1 - W1 = M1 * a
- T2 - W2 = M2 * a
- W3 = M3 * a
d. Next, substitute the expressions for the weight:
- T1 - M1 * g = M1 * a
- T2 - M2 * g = M2 * a
- M3 * g = M3 * a
e. Now, add equations 1 and 2, and solve for a:
- T1 - M1 * g + T2 - M2 * g = M1 * a + M2 * a
- T1 + T2 - (M1 + M2) * g = (M1 + M2) * a
- T1 + T2 - (26 kg + 38 kg) * 9.8 m/s^2 = (26 kg + 38 kg) * a
- T1 + T2 - 642.4 N = 64 kg * a (equation 4)
f. Lastly, substitute the expression for M3 * g into equation 4 and solve for a:
- (T1 + T2 - 642.4 N) + M3 * g = (M1 + M2 + M3) * a
- (T1 + T2 - 642.4 N) + (41 kg * 9.8 m/s^2) = (26 kg +38 kg + 41 kg) * a
- (T1 + T2 - 642.4 N) + 401.8 N = 105 kg * a
- T1 + T2 - 240.6 N = 105 kg * a
Now you have the equation to solve for the magnitude of acceleration (a).
2. Magnitude of Tension in each of the two strings:
To find the magnitude of tension in each of the two strings, we can go back to equations 1 and 2 from step b, and use the value of acceleration (a) obtained from step 1.
Substitute the value of acceleration into equations 1 and 2 to solve for T1 and T2.
- For M1 (26 kg):
T1 - M1 * g = M1 * a
- For M2 (38 kg):
T2 - M2 * g = M2 * a
Substitute the values of M1, M2, g, and a into the above equations and solve for T1 and T2. This will give you the magnitude of tension in each of the two strings.
To determine the magnitude of acceleration, we can analyze the system using Newton's second law. According to Newton's second law, the net force acting on an object is equal to its mass multiplied by its acceleration.
1. Determine the magnitude of acceleration:
Let's assume the direction of motion is to the right. We need to find the net force acting on the system.
a) For M1:
The force acting on M1 is the tension in the string connecting M1 and M2, which we'll call T1. Since there is no friction, the force acting on M1 is equal to T1. Therefore, the net force on M1 is T1 to the right.
b) For M2:
The force acting on M2 is the difference between the tension in the string connecting M1 and M2 (T1) and the tension in the string connecting M2 and M3 (T2). Therefore, the net force on M2 is (T1 - T2) to the right.
c) For M3:
The force acting on M3 is the tension in the string connecting M2 and M3, which we'll call T2. Therefore, the net force on M3 is T2 to the right.
Now, we can set up the equations using Newton's second law:
For M1:
T1 = M1 * a ----(1)
For M2:
(T1 - T2) = M2 * a ----(2)
For M3:
T2 = M3 * a ----(3)
To find the acceleration (a), we need to solve these equations simultaneously.
From equation (1):
T1 = 26kg * a
From equation (3):
T2 = 41kg * a
Substitute these values into equation (2):
(26kg * a - 41kg * a) = 38kg * a
Combine like terms:
-15kg * a = 38kg * a
Solve for a:
a = 0
Therefore, the magnitude of acceleration (a) is 0.
2. Determine the magnitude of tension in each of the two strings:
Since the acceleration is 0, the object is in equilibrium, and the tensions in the two strings are equal.
From equation (1):
T1 = 26kg * a
T1 = 0 * 26kg
T1 = 0N
From equation (3):
T2 = 41kg * a
T2 = 0 * 41kg
T2 = 0N
Therefore, the magnitude of the tension in each of the two strings is 0N.