Two blocks with masses M1 = 3.9 kg and M2 = 5.0 kg are connected with a massless string over two massless and frictionless pulleys, as shown in the Figure. One end of the string is connected to M1 while the other end is fixed. What is the acceleration of mass M2
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Two blocks of masses M1 and M2 are connected by a massless string that passes over a massless...
To find the acceleration of mass M2, we can first calculate the net force acting on the system.
1. Start by drawing a free body diagram for each mass.
For M1:
- There is a tension force T acting to the right.
- There is a gravitational force (weight) acting downward.
For M2:
- There is a tension force T acting to the left.
- There is a gravitational force (weight) acting downward.
2. Write the equations for the net force acting on each mass.
For M1:
- The net force acting on M1 is T - mg (where g is the acceleration due to gravity).
For M2:
- The net force acting on M2 is T - M2g.
3. Set up the equations of motion for each mass.
For M1:
- Using Newton's second law, F = ma, we can write T - mg = M1a.
For M2:
- Using Newton's second law, F = ma, we can write T - M2g = M2a.
4. Since the two masses are connected by a string, we know that the tension force is the same for both masses. Therefore, we can equate the two expressions for T.
T - mg = T - M2g
5. Simplify the equation.
-mg = -M2g
6. Cancel out the g term.
-m = -M2
7. Rearrange the equation to solve for M2.
M2 = m
8. Plug in the value of M2.
M2 = 5.0 kg
9. The acceleration of mass M2 is equal to the acceleration of mass M1, which we'll call a.
10. Substitute the value of M1 and M2 into one of the equations of motion to solve for a.
T - mg = M1a
T - (M1 + M2)g = M1a
T - (3.9 kg + 5.0 kg) * 9.8 m/s^2 = 3.9 kg * a
11. Solve for T.
T = (3.9 kg + 5.0 kg) * 9.8 m/s^2 + 3.9 kg * a
12. Substitute the value of T into the equation for M2.
(3.9 kg + 5.0 kg) * 9.8 m/s^2 + 3.9 kg * a - 5.0 kg * 9.8 m/s^2 = 5.0 kg * a
13. Simplify the equation.
9.8 kg * 9.8 m/s^2 + 3.9 kg * a - 5.0 kg * 9.8 m/s^2 = 5.0 kg * a
9.8 kg * 9.8 m/s^2 - 5.0 kg * 9.8 m/s^2 = 5.0 kg * a - 3.9 kg * a
9.8 kg * 9.8 m/s^2 - 5.0 kg * 9.8 m/s^2 = (5.0 kg - 3.9 kg) * a
14. Solve for a.
a = (9.8 kg * 9.8 m/s^2 - 5.0 kg * 9.8 m/s^2) / (5.0 kg - 3.9 kg)
a ≈ 9.18 m/s^2
Therefore, the acceleration of mass M2 is approximately 9.18 m/s^2.
To determine the acceleration of mass M2 in this system, we can use Newton's second law of motion.
First, let's identify the forces acting on each block:
For Mass M1:
- The force due to its own weight (mg1), where g is the acceleration due to gravity.
- The tension in the string (T) acting to the right.
For Mass M2:
- The force due to its own weight (mg2), where g is the acceleration due to gravity.
- The tension in the string (T) acting to the left.
Since the pulleys are frictionless, the tension in the string is the same throughout.
Now, let's write down the equations of motion for each block:
For Mass M1:
Sum of forces = Mass x Acceleration
mg1 - T = M1 * a (eq. 1)
For Mass M2:
Sum of forces = Mass x Acceleration
T - mg2 = M2 * a (eq. 2)
We can solve these two equations simultaneously to determine the acceleration (a).
First, solve equation 1 for T:
T = mg1 - M1 * a
Substitute this value of T into equation 2:
mg1 - M1 * a - mg2 = M2 * a
Rearrange the equation:
(mg1 - mg2) = (M1 + M2) * a
Finally, solve for the acceleration (a):
a = (mg1 - mg2) / (M1 + M2)
Now, plug in the given values:
m = 9.8 m/s^2 (acceleration due to gravity)
M1 = 3.9 kg
M2 = 5.0 kg
a = (3.9 kg * 9.8 m/s^2 - 5.0 kg * 9.8 m/s^2) / (3.9 kg + 5.0 kg)
Calculate the numerator:
(3.9 kg * 9.8 m/s^2 - 5.0 kg * 9.8 m/s^2) = -0.882 kg*m/s^2
Calculate the denominator:
(3.9 kg + 5.0 kg) = 8.9 kg
Finally, calculate the acceleration:
a = -0.882 kg*m/s^2 / 8.9 kg ≈ -0.099 m/s^2
The acceleration of mass M2 is approximately -0.099 m/s^2. Note that the negative sign indicates that it is moving in the opposite direction as expected from the initial setup.