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 = 9.0 kg, as shown in Figure P4.30. Find the acceleration of each object and the tension in the cable.
mass m1
mass m2
tension
Isnt the force pulling m2*g? And the mass being moved (m1+M2).
F=ma
3.5
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.
For object m1 (the one on the table):
Step 1: Identify the forces acting on m1. The only force acting on m1 is the tension in the cable, T.
Step 2: Apply Newton's second law to m1. The equation is:
m1 * a1 = T
Where a1 is the acceleration of m1.
For object m2 (the hanging object):
Step 1: Identify the forces acting on m2. The weight of m2 is acting downward, and the tension in the cable, T, is acting upward.
Step 2: Apply Newton's second law to m2. The equation is:
m2 * g - T = m2 * a2
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and a2 is the acceleration of m2.
Since the two objects are connected by the cable, the tension T will be the same for both objects.
Now we can solve the equations simultaneously to find the values of acceleration and tension.
From the equation for m1, we have:
m1 * a1 = T
From the equation for m2, we have:
m2 * g - T = m2 * a2
Since the tension T in the cable is the same for both objects, we can equate the two expressions for T:
T = m1 * a1 = m2 * g - T
Now, we can solve for a1:
a1 = (m2 * g) / (m1 + m2)
Substituting the given values:
m2 = 9.0 kg
m1 = 5.00 kg
g = 9.8 m/s^2
a1 = (9.0 kg * 9.8 m/s^2) / (5.00 kg + 9.0 kg) = 6.9 m/s^2
Now we can substitute this value of a1 into either equation to find T:
T = m1 * a1 = 5.00 kg * 6.9 m/s^2 = 34.5 N
So, the acceleration of each object is:
a1 = 6.9 m/s^2
a2 = -a1 (since m2 is hanging vertically)
And the tension in the cable is:
T = 34.5 N.
To find the acceleration of each object and the tension in the cable, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's denote the acceleration of both objects as a, the tension in the cable as T, and the gravitational acceleration as g.
For the object with mass m1:
The only force acting on it is the tension in the cable. Since the table is frictionless, there is no friction force. According to Newton's second law, we have:
T - m1 * g = m1 * a -- (Equation 1)
For the hanging object with mass m2:
The gravitational force is acting downwards, and the tension in the cable is acting upwards. Using Newton's second law, we have:
m2 * g - T = m2 * a -- (Equation 2)
Now, our goal is to solve these two equations simultaneously to find the acceleration of each object and the tension in the cable.
First, let's eliminate T from the equations. Rearrange Equation 2 as:
T = m2 * g - m2 * a
Substituting this into Equation 1, we have:
m2 * g - m2 * a - m1 * g = m1 * a
Simplifying the equation by combining like terms, we get:
(m1 + m2) * a = m2 * g - m1 * g
Finally, divide both sides of the equation by (m1 + m2) to solve for a:
a = (m2 * g - m1 * g) / (m1 + m2)
This is the acceleration of both objects.
To find the tension in the cable, substitute the value of a back into Equation 2:
T = m2 * g - m2 * a
Now, you can substitute the given values of m1, m2, and g into the equations to find the acceleration and tension.