4 kg mass is placed on the smooth horizontal tabke.two light inextansible strings are connected to it and two masse 2kg and 5kg are attached to the free ends of the strings. Find the accelaration of each objects and the tension of the strings
To find the acceleration of each object and the tension in the strings, we can use Newton's second law of motion.
First, let's consider the 4 kg mass on the table. Since it is on a smooth horizontal surface, there is no friction acting on it. The only external force acting on it is the tension in the strings.
Let's denote the tension in the string connected to the 2 kg mass as T1, and the tension in the string connected to the 5 kg mass as T2.
Now, applying Newton's second law to the 4 kg mass:
Sum of forces = mass × acceleration
(T1 + T2) - (4 kg × g) = 4 kg × a
(T1 + T2) = 4 kg × a + (4 kg × g) -----(Equation 1)
Next, let's consider the 2 kg mass:
Sum of forces = mass × acceleration
T1 - (2 kg × g) = 2 kg × a1
T1 = 2 kg × a1 + (2 kg × g) -----(Equation 2)
Similarly, let's consider the 5 kg mass:
Sum of forces = mass × acceleration
T2 - (5 kg × g) = 5 kg × a2
T2 = 5 kg × a2 + (5 kg × g) -----(Equation 3)
We also know that the strings are light and inextensible, which means the tension in both strings will be the same:
T1 = T2 -----(Equation 4)
Now, we have four equations (Equation 1, 2, 3, and 4) with four unknowns (T1, T2, a1, and a2). By solving these equations simultaneously, we can find the values of the tensions and accelerations.
Let's substitute Equation 4 into Equations 2 and 3:
2 kg × a1 + (2 kg × g) = 5 kg × a2 + (5 kg × g) -----(Equation 5)
Subtracting Equation 5 from Equation 1:
4 kg × a + (4 kg × g) - (2 kg × a1 + (2 kg × g)) = 4 kg × a
2 kg × a - 2 kg × a1 = 2 kg × g - (4 kg × g)
2 kg × (a - a1) = 2 kg × (g - 2g)
a - a1 = -g
Now, let's subtract Equation 3 from Equation 1:
4 kg × a + (4 kg × g) - (5 kg × a2 + (5 kg × g)) = 4 kg × a
- kg × a - kg × a2 = - kg × g
a + a2 = g
We now have two equations with two unknowns (a1 and a2). By solving this system of equations, we can find the accelerations of the 2 kg and 5 kg masses.
Once we have the values of a1 and a2, we can substitute them back into Equations 2 and 3 to find the tensions T1 and T2.
Please note that the value of "g" is the acceleration due to gravity, which is approximately 9.8 m/s^2.
To find the acceleration of each object and the tension in the strings, we can use Newton's second law of motion. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Let's denote the 4 kg mass as M, the 2 kg mass as m1, and the 5 kg mass as m2.
Step 1: Calculate the net force acting on each object:
- For the 4 kg mass (M), there is no net force acting on it since it is placed on a smooth horizontal table. Therefore, the net force acting on M is zero (0 N).
- For the 2 kg mass (m1), the tension force is acting to the right, and there is no other force acting on it horizontally. Therefore, the net force acting on m1 is equal to the tension force, denoted as T.
- For the 5 kg mass (m2), the tension force is acting to the left, and there is no other force acting on it horizontally. Therefore, the net force acting on m2 is equal to the tension force, denoted as T.
Step 2: Set up the equations of motion for each object:
For M (4 kg):
Since the net force acting on M is zero, the equation becomes:
0 N = M * aM, where aM is the acceleration of M.
For m1 (2 kg):
Using the equation F = m * a, where m is the mass and a is the acceleration, we have:
T - 0 N = m1 * a1
For m2 (5 kg):
Using the equation F = m * a, where m is the mass and a is the acceleration, we have:
0 N - T = m2 * a2
Step 3: Solve the equations simultanously:
Since we have two equations with two unknowns (a1 and a2), we can solve them simultaneously.
From the equation for M (4 kg), we know that the acceleration of M is zero, aM = 0.
From the equations for m1 (2 kg) and m2 (5 kg), we have:
T - 0 N = m1 * a1
0 N - T = m2 * a2
Simplifying the equations further:
T = m1 * a1
T = m2 * a2
Since T = T (the tension in both strings is the same), we can set the two equations equal to each other:
m1 * a1 = m2 * a2
Substituting the given mass values, we get:
2 kg * a1 = 5 kg * a2
Dividing both sides by 2 kg:
a1 = (5 kg / 2 kg) * a2
a1 = 2.5 * a2
Step 4: Calculate the values of a1, a2, and T:
Since there are multiple possible values that satisfy the equation, we can choose any arbitrary value for a2. For simplicity, let's assume a2 = 1 m/s^2.
Using the equation a1 = 2.5 * a2, we can substitute the value of a2 and solve for a1:
a1 = 2.5 * 1 m/s^2
a1 = 2.5 m/s^2
Finally, we can substitute the values of a1 and a2 into the equation T = m1 * a1 or T = m2 * a2 to calculate the tension in the strings:
T = 2 kg * 2.5 m/s^2
T = 5 N
So, the acceleration of the 2 kg mass (m1) is 2.5 m/s^2, the acceleration of the 5 kg mass (m2) is 1 m/s^2, and the tension in the strings is 5 N.