In this problem they show a diagram of a pulley system with an inclined plane (25 degree incline). There is a box called m1=47kg resting on the incline plane and m1=35kg hanging on the vertical line. The question reads: The coefficient of friction between m1 and the surfvace of the inclined plane are Ustatic=0.42 and Ukinetic=0.19.
a)if the masses are held in place and then released , will they start to move? ------> I know the answer is NO.
b)how much mass would you have to add to m2 to cause the masses to begin to move? ----> The answer should be 2.8kg but I don't know how the book got this answer.
c)If you said no to a) and added the mass that you calculated in b), what would be the acceleration of the masses? ---> answer given in the back of textbook is 5.7 m/s^2
ANY help with this problem would be greatly appreciated!! THANK YOU SO MUCH in advance :)
W1 = m1g = 47kg * 9.8N/kg = 461 N
W2 = m2g = 35kg * 9.8N/kg = 343 N.
F1 = 461N @ 25 Deg.
Fp = 461*sin25 = 195 N.=Force parallel
to the incline.
Fv = 461*cos25 = 418 N. = Force perpendicular to incline.
Fn = F2 - Fp - Fs.
Fn = 343 - 195 - 0.42*418 = 0.
Fn = 343 - 195 - 176 = -28 N.
a.No, the masses would not move, because the opposing force is 28 N greater than F2. Therefore we must add
28 N. to F2:
b. mg = 28 N.
m = 28/g = 28 / 9.8 = 2.86 kg added.
thank you very much! and to find the acceleration would it just be Fnet=ma and solve for a... Fnet=Fg-Ff?
To solve this problem, we can start by analyzing the forces acting on the system. Let's assume that the positive x-direction is parallel to the inclined plane and the positive y-direction is perpendicular to the inclined plane.
a) To determine if the masses will start to move, we need to consider the forces on m1. There are three forces acting on m1: its weight (mg), the normal force from the inclined plane (N), and the force of friction (f). Since the system is at rest in this case, the friction force opposes the impending motion and is equal to the maximum static friction force (f_s = U_s * N).
The weight component along the inclined plane is m1 * g * sin(theta), and the weight component perpendicular to the inclined plane is m1 * g * cos(theta). Since the system is at rest, the friction force (f) must be equal in magnitude to the weight component along the incline, so f = m1 * g * sin(theta).
If f is less than or equal to the maximum static friction force (f_s), the system will remain at rest. Therefore, we need to compare f_s and f to determine if the masses will start to move.
f_s = U_s * N = U_s * m1 * g * cos(theta)
Substituting the given values, we have:
f_s = 0.42 * 47 * 9.8 * cos(25) = 182.961 N
f = m1 * g * sin(theta) = 47 * 9.8 * sin(25) = 197.36 N
Since f > f_s, the system will start to move. However, the question asks if the masses will start to move if they are held in place and then released, which means considering any external forces holding the masses in place. Since these external forces are not specified, we can assume there are no external forces holding the masses in place, and the answer is NO.
b) To determine how much mass needs to be added to m2 to cause the masses to begin to move, we need to consider the forces again. In this case, the force of friction will change from static friction (maximum value) to kinetic friction (f_k = U_k * N).
When the masses are just about to start moving, the friction force is equal to the weight component along the inclined plane (m1 * g * sin(theta)). We can express this as:
f_k = U_k * N = U_k * m1 * g * cos(theta)
Substituting the given values, we have:
U_k * m1 * g * cos(theta) = m1 * g * sin(theta)
dividing both sides by g * m1 * cos(theta):
U_k = sin(theta) / cos(theta) = tan(theta)
Substituting the value of theta (25 degrees), we find:
U_k = tan(25) ≈ 0.46631
Now, we can solve for m2 using the formula for the friction force (f_k) and the weight of m2 (m2 * g):
f_k = m2 * g = U_k * m1 * g * cos(theta)
m2 * g = U_k * m1 * g * cos(theta)
m2 = U_k * m1 * cos(theta)
Substituting the given values, we have:
m2 ≈ 0.46631 * 47 * cos(25) ≈ 2.8 kg
Therefore, you would need to add approximately 2.8 kg to m2 to cause the masses to begin to move.
c) If you said "no" to part (a) and added the mass calculated in part (b), we can determine the acceleration of the masses. Once the masses start to move, they experience a constant friction force equal to the kinetic friction force (f_k) between m1 and the inclined plane.
To find the acceleration (a), we can use Newton's second law:
net force = m_eff * a
The net force on the system is the force of gravity acting on m2 (m2 * g) minus the kinetic friction force (f_k):
net force = m2 * g - f_k
m_eff is the effective mass of the system and is given by:
m_eff = m1 + m2
Substituting the values, we have:
m_eff * a = m2 * g - f_k
(m1 + m2) * a = m2 * g - U_k * m1 * g * cos(theta)
Simplifying, we find:
a = (m2 * g - U_k * m1 * g * cos(theta)) / (m1 + m2)
Substituting the given values, we have:
a = (2.8 * 9.8 - 0.46631 * 47 * 9.8 * cos(25)) / (47 + 2.8)
a ≈ 5.7 m/s²
Therefore, the acceleration of the masses would be approximately 5.7 m/s².