As shown in the figure, two masses, m 1 = 3.41 kg and m 2 = 4.85 kg, are on a fr
ID: 1767720 • Letter: A
Question
As shown in the figure, two masses, m1 = 3.41 kg and m2 = 4.85 kg, are on a frictionless tabletop and mass m3 = 8.91 kg is hanging from m1. The coefficients of static and kinetic friction between m1 and m2 are 0.537 and 0.437, respectively.
a.) What are the accelerations of m1 and m2?
a1 = ________
a2 = ________
b.) What is the tension in the string between m1 and m3?
As shown in the figure, two masses, m1 = 3.41 kg and m2 = 4.85 kg, are on a frictionless tabletop and mass m3 = 8.91 kg is hanging from m1. The coefficients of static and kinetic friction between m1 and m2 are 0.537 and 0.437, respectively. a.) What are the accelerations of m1 and m2? a1 = a2 = b.) What is the tension in the string between m1 and m3?Explanation / Answer
Let's use the following notations:
- F is the horizontal force of friction between m1 and m2.
- a is the acceleration of m2.
- b is the acceleration of m1 and m3
- T is the tension of the string.
- g = 9.80665 m/s^2 is the gravitational field (see first link below).
- u = 0.437 is the coefficient of kinetic friction.
m2 a = F
m1 b = T - F
m3 b = m3 g - T
Therefore:
T = m3 (g - b)
F = m3 (g - b) - m1 b = m2 a
Under the "static regime" where m1 and m2 are at rest with respect to each other,
we have a=b so that the last equation above yields
m3 g = (m1+m2+m3) a
F = m2 a = g m2 m3 / (m1+m2+m3)
The coefficient of static friction must exceed the ratio of F to the normal force:
F / (m1 g) = m2 m3 / [m1 (m1+m2+m3)] < 0.537
With the values given, the left-hand-side is 0.738
Therefore, the above inequality does not hold.
This means that we are dealing with the kinetic regime where a and b need not
be equal. In addition to the above, the following equality holds:
F = u m1 g [where u = 0.437]
Therefore, the acceleration of m2 is:
a = F/m2 = g u m1/m2 = 0.282 g = 3.03 m/s^2
The acceleration b of m1 (and m3) is obtained from the original equation:
F = g u m1 = m3 (g-b) - m1 b
b = g (m3 - u m1) / (m1 + m3) = 5.9 m/s^2
Finally, the tension of the string is:
T = m3 (g - b) = g m3 [m1 + m3 - m3 + u m1 ] / (m1+m3)
T = g (1+u) m1 m3 / (m1+m3) = 34.73 N
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