Three objects with masses

m1= 36.5 kg
m2= 19.2 kg
m3= 12.5 kg
Are hanging from massless ropes that run over a frictionless pulleys. If the system is released from rest, what is the magnitude of the acceleration in m/s^2 of mass m1?

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m2 m1 m3

To find the magnitude of the acceleration of mass m1, we need to analyze the forces acting on it.

1. Start by drawing a free-body diagram for mass m1, indicating all the forces acting on it. In this case, we have the force of gravity (mg) pulling it downward and the tension force (T) in the rope pulling it upward.

T

m1↓

mg

2. The force of gravity on mass m1 can be calculated using the formula F = mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and m1 is the mass of m1:

F_gravity = m1 * g

F_gravity = 36.5 kg * 9.8 m/s^2

F_gravity = 357.7 N

3. Since the system is released from rest, we know that the tension force in the rope will cause an acceleration. Let's call this acceleration a.

4. Now, consider mass m2. The tension force in the rope will cause mass m2 to accelerate downward. This force can be calculated using Newton's second law of motion:

F = m2 * a

T = m2 * a

T = 19.2 kg * a

5. Likewise, for mass m3, the tension force in the rope will cause mass m3 to accelerate downward:

T = m3 * a

T = 12.5 kg * a

6. Therefore, the total tension force acting on mass m1 is the sum of the tension forces from masses m2 and m3:

T_total = T_m2 + T_m3

= 19.2 kg * a + 12.5 kg * a

= (19.2 + 12.5) kg * a

= 31.7 kg * a

7. The net force acting on mass m1 is the difference between the force of gravity and the total tension force:

F_net = F_gravity - T_total

= 357.7 N - (31.7 kg * a)

8. We know that Newton's second law of motion states that the net force acting on an object is equal to its mass multiplied by its acceleration:

F_net = m1 * a

9. Equating the net force expressions from steps 7 and 8:

357.7 N - (31.7 kg * a) = 36.5 kg * a

10. Solve the equation for the acceleration (a):

357.7 N - 31.7 kg * a = 36.5 kg * a

357.7 N = (36.5 kg + 31.7 kg) * a

357.7 N = 68.2 kg * a

a = 357.7 N / 68.2 kg

a ≈ 5.25 m/s^2

Therefore, the magnitude of the acceleration of mass m1 is approximately 5.25 m/s^2.