Objects of masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley as in the figure below. The object m1 is held at rest on the floor, and m2 rests on a fixed incline of θ = 38.0°. The objects are released from rest, and m2 slides 1.60 m down the slope of the incline in 3.90 s.
(a) Determine the acceleration of each object. (Enter the magnitude only.)
m/s2
(b) Determine the tension in the string. (Enter the magnitude only.)
N
(c) Determine the coefficient of kinetic friction between m2 and the incline.
To solve this problem, we can use Newton's laws of motion and apply them to each object separately. Let's start with the inclined plane and object m2.
(a) Determine the acceleration of object m2:
We can use the equation for acceleration in terms of distance and time:
a = 2 * (s - u * t) / t^2
where s is the distance traveled, u is the initial velocity, and t is the time interval.
Given that s = 1.60 m and t = 3.90 s, we need to find u, the initial velocity. Since object m2 is released from rest, its initial velocity is 0 m/s.
Plugging the values into the formula:
a = 2 * (1.60 m - 0 * 3.90 s) / (3.90 s)^2
a = 2 * 1.60 m / 15.21 s^2
a = 0.21 m/s^2
Therefore, the acceleration of object m2 is 0.21 m/s^2.
Now, let's move on to object m1.
(a) Determine the acceleration of object m1:
Since object m1 is held at rest on the floor, it does not move. Therefore, its acceleration is 0 m/s^2.
(b) Determine the tension in the string:
To find the tension in the string, we can use the equation:
tension = m2 * (g - a)
where g is the acceleration due to gravity (9.8 m/s^2).
Plugging in the values:
tension = 9.00 kg * (9.8 m/s^2 - 0.21 m/s^2)
tension = 9.00 kg * 9.59 m/s^2
tension = 86.31 N
Therefore, the tension in the string is 86.31 N.
(c) Determine the coefficient of kinetic friction between m2 and the incline:
To find the coefficient of kinetic friction, we can use the equation:
frictional force = coefficient of friction * normal force
The normal force on object m2 is equal to its weight:
normal force = m2 * g
normal force = 9.00 kg * 9.8 m/s^2
normal force = 88.2 N
The frictional force is the force that opposes the motion of object m2 down the incline:
frictional force = m2 * a
frictional force = 9.00 kg * 0.21 m/s^2
frictional force = 1.89 N
Plugging the values into the equation:
1.89 N = coefficient of friction * 88.2 N
Solving for the coefficient of friction:
coefficient of friction = 1.89 N / 88.2 N
coefficient of friction ≈ 0.0214
Therefore, the coefficient of kinetic friction between object m2 and the incline is approximately 0.0214.