A 1.2kg object is placed on a frictionless table and is attached to a hanging mass using a light string over a frictionless pulley in a fletcher's trolley arrangement. The 1.20kg object accelerates along the table at 2.31m/s while the hanging object accelerates down at the same rate. Determine the mass of the hanging object.
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To solve this problem, we can use Newton's second law of motion and the concept of net force.
1. First, let's calculate the net force acting on the 1.2kg object along the table. We know its mass (m = 1.2 kg) and acceleration (a = 2.31 m/s^2). Using Newton's second law, we can calculate the net force (F_net) as:
F_net = m * a
F_net = 1.2 kg * 2.31 m/s^2
F_net = 2.772 N
2. Since the table is frictionless, the force pulling the 1.2kg object along the table is due to the tension in the string. This force is also acting on the hanging mass. Therefore, the net force acting on both objects is the same.
3. Now, let's calculate the weight of the 1.2kg object (force due to gravity acting on the object) using the equation:
Weight = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Weight = 1.2 kg * 9.8 m/s^2
Weight = 11.76 N
4. Since the hanging mass is accelerating at the same rate as the 1.2kg object, the net force on the hanging mass is also 2.772 N.
5. The net force on the hanging mass is the difference between the weight of the hanging mass and the tension in the string. Mathematically, this can be expressed as:
F_net = Tension - Weight_hanging_mass
where Weight_hanging_mass is the force due to gravity acting on the hanging mass.
6. Rearranging the equation, we get:
Tension = F_net + Weight_hanging_mass
Tension = 2.772 N + Weight_hanging_mass
7. Now, let's find the weight of the hanging mass. We can rewrite the weight equation as:
Weight_hanging_mass = m_hanging_mass * g
where m_hanging_mass is the mass of the hanging object.
Tension = 2.772 N + m_hanging_mass * g
8. We have one more piece of information: the acceleration along the table is the same as the acceleration of the hanging mass. Since the hanging mass is pulled down, its acceleration is negative. Therefore, we can say:
m_hanging_mass * g = (-1) * (1.2 kg * 2.31 m/s^2)
9. Substituting this into the previous equation, we get:
Tension = 2.772 N + (-1) * (1.2 kg * 2.31 m/s^2)
Tension = 2.772 N - 2.682 N
Tension = 0.09 N
10. Now that we have the tension in the string, we can go back to the equation for the net force on the hanging mass:
F_net = Tension - Weight_hanging_mass
2.772 N = 0.09 N - Weight_hanging_mass
11. Rearranging the equation, we get:
Weight_hanging_mass = 0.09 N - 2.772 N
Weight_hanging_mass = -2.682 N
12. Since weight is a positive quantity, we can change the sign of the force acting on the hanging mass to make it positive:
Weight_hanging_mass = 2.682 N
13. Finally, we can find the mass of the hanging object by rearranging the weight equation:
Weight_hanging_mass = m_hanging_mass * g
2.682 N = m_hanging_mass * 9.8 m/s^2
m_hanging_mass = 2.682 N / 9.8 m/s^2
m_hanging_mass ≈ 0.274 kg
Therefore, the mass of the hanging object is approximately 0.274 kg.
To determine the mass of the hanging object, we can start by analyzing the forces acting on the system.
Let's denote the mass of the hanging object as M.
For the object on the table (with mass 1.2 kg), we have:
1. The force of gravity acting on it (m1 * g), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The tension in the string pulling it in the direction of the hanging mass (T).
Since the object on the table is accelerating at 2.31 m/s^2, there must be an unbalanced force acting on it. This force is the tension in the string. Therefore, we can write the equation for the object on the table as:
T = m1 * a
For the hanging object, we have:
1. The force of gravity acting on it (M * g).
2. The tension in the string pulling it in the opposite direction (-T).
Since the hanging object is also accelerating at 2.31 m/s^2, the net force acting on it is zero, so we can write the equation for the hanging object as:
M * g - T = 0
Now, we can substitute the expression for T from the first equation into the second equation:
M * g - m1 * a = 0
Rearranging the equation, we get:
M = (m1 * a) / g
Plugging in the known values:
m1 = 1.2 kg
a = 2.31 m/s^2
g = 9.8 m/s^2
M = (1.2 kg * 2.31 m/s^2) / 9.8 m/s^2
Calculating this expression gives us the mass of the hanging object, M = 0.284 kg (rounded to three decimal places).
Therefore, the mass of the hanging object is approximately 0.284 kg.