An object with mass m1=5.00 kg rests on a frictionless horizontal table and is connected to a cable that passes over a pulley and is then fastened to a hanging object with mass m2= 10.0 kg. Find the acceleration of each object and the tension in the cable.
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An object with mass m1 =5.00kg rests on frictionless horizontal table and is connected to cable that passes over a pulley and is then fastened to a hanging object with mass m2=10kg.find the acceleration of each object and then tension on a string.
To find the acceleration of each object and the tension in the cable, we can use Newton's second law of motion and consider the forces acting on each object separately.
1. For the hanging object (mass m2 = 10.0 kg):
- The gravitational force acting on it is given by F_gravity = m2 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The tension in the cable is the force acting on the hanging object in the upward direction.
- Using Newton's second law, we have F_net = m2 * a, where a is the acceleration of the hanging object.
- Since the hanging object is accelerating upwards, we can write F_net = F_gravity - Tension.
- Substituting the values, m2 * a = m2 * g - Tension.
2. For the object on the table (mass m1 = 5.00 kg):
- The tension in the cable is the force acting on the object in the horizontal direction.
- Using Newton's second law, we have F_net = m1 * a, where a is the acceleration of the object.
- Since the object is on a frictionless table, the only force acting on it is the tension in the cable.
- Substituting the values, m1 * a = Tension.
Now, we can solve the equations simultaneously.
Equate the expressions for tension from both equations:
m1 * a = m2 * g - Tension
Substituting m1 * a = Tension,
Tension = m2 * g - m1 * a
Next, equate the expressions for acceleration from both equations:
m2 * a = m2 * g - Tension
Substituting Tension = m1 * a,
m2 * a = m2 * g - m1 * a
Now, solve for acceleration (a):
m2 * a + m1 * a = m2 * g
(a * m2 + a * m1) = m2 * g
a * (m2 + m1) = m2 * g
a = (m2 * g) / (m2 + m1)
Substituting the values:
a = (10.0 kg * 9.8 m/s^2) / (10.0 kg + 5.00 kg)
a = 9.8 m/s^2 / 15.0 kg
a = 0.653 m/s^2
Therefore, the acceleration of each object is 0.653 m/s^2.
To find the tension in the cable, we can substitute this acceleration value back into the equation for tension:
Tension = m2 * g - m1 * a
Tension = (10.0 kg * 9.8 m/s^2) - (5.00 kg * 0.653 m/s^2)
Tension = 98.0 N - 3.27 N
Tension = 94.73 N
Therefore, the tension in the cable is 94.73 N.
To find the acceleration of each object and the tension in the cable, we can apply Newton's second law of motion to both objects.
Let's assume that the hanging object (m2) moves downwards and the object on the table (m1) stays on the table.
For the object hanging vertically (m2):
- The weight of m2 acts downwards, given by F2 = m2 * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
- The tension in the cable acts upwards.
Using Newton's second law, we have:
m2 * a = F2 - T
For the object on the table (m1):
- The tension in the cable acts to the right.
- The weight of m1 is balanced by the normal force from the table, which cancels out the force of gravity.
Using Newton's second law, we have:
m1 * a = T
Since the objects are connected by the cable, both objects will have the same acceleration, denoted as 'a'.
We can now solve the equations simultaneously to find the values of 'a' and 'T'.
Substituting the second equation into the first equation, we get:
m2 * a = m1 * a + T
Rearranging, we have:
T = (m2 - m1) * a
We also know that the weight of m2 (F2) is given by m2 * g.
Substituting m2 * g for F2, we can further simplify the equation to find 'a':
(m2 - m1) * a = m2 * g
Now, we can solve for 'a':
a = (m2 * g) / (m2 - m1)
Once you substitute the given values for m1 (5.00 kg) and m2 (10.0 kg), and use the value of g (9.8 m/s²), you can calculate the acceleration ('a').
To find the tension in the cable, substitute the value of 'a' back into the equation:
T = (m2 - m1) * a
Once you substitute the values of m1, m2, and the calculated value of 'a', you can find the tension ('T') in the cable.