There are two objects of masses M and m. The horizontal surface allows for frictionless motion. The string ties to the two objects is massless and passes over a massless pulley that rotates without friction. If M=4.23 kg and m=1.24 kg, what is the magnitude of the acceleration of the sliding object?
To determine the magnitude of the acceleration of the sliding object in this system, we can apply Newton's second law of motion.
First, let's define the forces acting on each object:
1. Object of mass M:
- Force due to gravity (weight) = M * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2)
2. Object of mass m:
- Force due to gravity (weight) = m * g
Since the objects are connected by a massless string passing over a massless pulley, the tension in the string will be the same for both objects.
Now, let's analyze the motion of the system. We'll assume that the acceleration of the sliding object is positive to the right:
1. Object of mass M:
- Net force = Tension - M * g = M * a, where a is the acceleration of the sliding object
- Tension = M * g + M * a
2. Object of mass m:
- Net force = m * g - Tension = m * a
- Tension = m * g - m * a
Since the tension in the string is the same for both objects, we can equate the two expressions for tension:
M * g + M * a = m * g - m * a
Now, we can solve this equation for the acceleration (a):
M * a + m * a = m * g - M * g
(a * (M + m)) = (m - M) * g
a = (m - M) * g / (M + m)
Plugging in the known values:
M = 4.23 kg
m = 1.24 kg
g = 9.8 m/s^2
a = (1.24 kg - 4.23 kg) * 9.8 m/s^2 / (4.23 kg + 1.24 kg)
a ≈ -2.683 m/s^2 (approximated to three decimal places)
Therefore, the magnitude of the acceleration of the sliding object is approximately 2.683 m/s^2.