Two packing crates of masses 7.93 kg and 5.13 kg are connected by a light string that passes over a frictionless pulley as in the figure below. The 5.13 kg crate lies on a smooth incline of angle 40.0°.
a) Find the acceleration of the 5.13 kg crate
B)Find the tension in the string
m2•a=T - m2•g•sinα => T= m2•a + m2•g•sinα
m1•a=m1•g – T= m1•g - m2•a - m2•g•sinα.
a=g•(m1-m2•sinα)/(m1+m2) = 9.8•(7.93-5.13• sin40°)/(7.93+5.13) =
=3.48 m/s²,
T= m1(g-a)= 7.93•(9.8-3.48) = 50.15 N.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration. We will break down the problem into two parts - finding the acceleration of the 5.13 kg crate and finding the tension in the string.
a) Finding the acceleration of the 5.13 kg crate:
Since the 5.13 kg crate lies on a smooth incline, the gravitational force acting on it can be resolved into two components - one parallel to the incline and one perpendicular to it.
The component parallel to the incline is given by F_parallel = m * g * sin(theta), where m is the mass of the crate, g is the acceleration due to gravity, and theta is the angle of the incline.
The net force acting on the crate parallel to the incline can be given by F_net = m * a, where m is the mass of the crate and a is the acceleration of the crate.
Therefore, we can set up the equation: m * a = m * g * sin(theta)
Rearranging the equation, we get: a = g * sin(theta)
b) Finding the tension in the string:
The tension in the string can be found by considering the system as a whole. The net force acting on the system is the difference between the gravitational force of each crate. Since the string is assumed to be light, we can assume that the tension in the string is the same on both sides.
The net force acting on the system is given by: Net force = (m1 - m2) * g, where m1 and m2 are the masses of the crates, and g is the acceleration due to gravity.
Therefore, the tension in the string is equal to the net force acting on the system.
Tension = (m1 - m2) * g
Using the given masses and values, substitute the values into the equations to find the acceleration and tension.