Two masses of 8 kg and 12 kg are tied to each end f a string passing over a massless and frictionless pulley. find the resulting accelration of each mass.
To find the resulting acceleration of each mass, we'll first need to determine the net force acting on each mass.
Let's label the mass of 8 kg as m1 and the mass of 12 kg as m2.
1. Determine the force due to gravity acting on each mass:
- Force of gravity on m1 (F1) = m1 * g = 8 kg * 9.8 m/s^2 = 78.4 N (upwards)
- Force of gravity on m2 (F2) = m2 * g = 12 kg * 9.8 m/s^2 = 117.6 N (downwards)
2. Determine the net force acting on each mass:
- Net force on m1 (F_net1) = F2 - F1 = 117.6 N - 78.4 N = 39.2 N (downwards)
- Net force on m2 (F_net2) = F1 - F2 = 78.4 N - 117.6 N = -39.2 N (upwards)
3. Apply Newton's second law (F = m * a) to find the resulting acceleration:
- F_net1 = m1 * a1 => 39.2 N = 8 kg * a1 => a1 = 39.2 N / 8 kg = 4.9 m/s^2 (downwards)
- F_net2 = m2 * a2 => -39.2 N = 12 kg * a2 => a2 = -39.2 N / 12 kg = -3.27 m/s^2 (upwards)
Therefore, the resulting acceleration of the 8 kg mass is 4.9 m/s^2 (downwards), and the resulting acceleration of the 12 kg mass is 3.27 m/s^2 (upwards).
To find the resulting acceleration of each mass in this system, we can apply Newton's laws of motion.
Let's denote the acceleration of the system as 'a'.
For the 8 kg mass (m1):
- The force acting on m1 is the tension in the string (T).
- The gravitational force acting on m1 is given by the equation F1 = m1 * g, where g represents the acceleration due to gravity.
- As the string is inextensible, the tension in the string (T) will be the same throughout its length.
According to Newton's second law, the net force acting on an object is equal to the product of its mass and acceleration. Therefore, we can write the equation of motion for m1 as:
m1 * a = T - F1
For the 12 kg mass (m2):
- The gravitational force acting on m2 is given by the equation F2 = m2 * g.
According to Newton's second law, the net force acting on an object is equal to the product of its mass and acceleration. Therefore, we can write the equation of motion for m2 as:
m2 * a = F2 - T
Since both masses are connected by a string passing over a massless and frictionless pulley, the tension in the string will be the same for both masses. Therefore, T = T.
Now, let's set up the equations and solve them simultaneously to find the resulting acceleration (a):
For m1:
m1 * a = T - F1
8 kg * a = T - (8 kg * g)
For m2:
m2 * a = F2 - T
12 kg * a = (12 kg * g) - T
Since T = T, we can equate the two equations and solve for 'a':
8 kg * a = (12 kg * g) - T
8 kg * a = (12 kg * g) - (8 kg * g)
8 kg * a = 4 kg * g
a = (4 kg * g) / 8 kg
a = 0.5 g
Therefore, the resulting acceleration of each mass in the system is 0.5 times the acceleration due to gravity (g).