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, as shown in the figure below. Find the acceleration of each object and the tension in the cable.
The pulling force is m2*g
the total mass accelerating is m2+m1
Pulling force=totalmass* acceleration
Tension: It is pulling the table force
Tension=m1*a
To find the acceleration of each object and the tension in the cable, we can utilize Newton's second law of motion and consider the forces acting on each object.
1. Let's start by considering the object with mass m1 on the table:
- The only force acting on this object is its weight, mg1, directed downwards.
- Since the table is frictionless, there is no frictional force.
- According to Newton's second law, the net force on an object is equal to the product of its mass and acceleration: ΣF = m1 * a.
2. For the hanging object with mass m2:
- The weight of this object, mg2, is acting downwards.
- We also have the tension force in the cable, T, acting upwards.
- Applying Newton's second law, ΣF = m2 * a, for the hanging object.
3. Additionally, both objects are connected by the cable, so the tension in the cable is the same for both objects.
Now, based on the above analysis, we can write down the equations:
1. For the object on the table:
ΣF = m1 * a
mg1 = m1 * a (since there are no other forces)
a = g1 (where g1 represents the acceleration due to gravity, approximately 9.8 m/s²)
2. For the hanging object:
ΣF = m2 * a
T - mg2 = m2 * a (considering upward forces positive)
T = m2 * a + mg2
Now, we can substitute the expression for acceleration (a) from equation 1 into equation 2:
T = m2 * g1 + m2 * a
T = m2 * (g1 + a)
Since both objects are connected by the same cable, T is the tension in the cable for both objects.
Thus, the tension in the cable is T = m2 * (g1 + a), and the acceleration of each object is a = g1.
To get the numerical values, substitute the known values of m1, m2, and g1 into the equations.