Two masses, 5 kg and 8 kg, are connected by a string passing over a pulley. Determine the acceleration of the masses and the tension in the string.
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We can start by drawing a free-body diagram for each mass.
For the 5 kg mass:
- There is a tension force acting upwards.
- There is a gravitational force acting downwards with a magnitude of 5 kg * 9.8 m/s^2 = 49 N (using g = 9.8 m/s^2).
For the 8 kg mass:
- There is a tension force acting downwards.
- There is a gravitational force acting downwards with a magnitude of 8 kg * 9.8 m/s^2 = 78.4 N.
We can now apply Newton's second law of motion to each mass:
For the 5 kg mass:
- ΣF = ma
- T - 49 N = 5 kg * a
- T = 5 kg * a + 49 N
For the 8 kg mass:
- ΣF = ma
- 78.4 N - T = 8 kg * a
- T = 78.4 N - 8 kg * a
Since the two masses are connected by the same string passing over a pulley, the tension in the string is the same for both masses, so we can set the two expressions for T equal to each other:
5 kg * a + 49 N = 78.4 N - 8 kg * a
Rearranging the equation:
13 kg * a = 29.4 N
Simplifying the equation:
a = 29.4 N / 13 kg = 2.26 m/s^2
The acceleration of the masses is 2.26 m/s^2.
To find the tension in the string, we can substitute the value of a into one of the expressions for T:
T = 5 kg * a + 49 N
T = 5 kg * 2.26 m/s^2 + 49 N
T = 11.3 N + 49 N
T = 60.3 N
The tension in the string is 60.3 N.
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