A 10-lb block B rests as shown on a 20-lb bracket A. The coefficients of friction are μs = 0.30 and μk = 0.25 between block B and bracket A, and there is no friction in the pulley or between the bracket and the horizontal surface. (a) Determine the maximum weight of block C if block B is not to slide on bracket A. (b) If the weight of block C is 10 percent larger than the answer found in a, determine the accelerations of A, B, and C.
Fig. P12.31
(a)
Find the maximum weight of block C if block B is not to slide on bracket A.
Answer to Problem 12.31P
The maximum weight of block C if block B is not to slide on bracket A is
Explanation of Solution
Given information:
The weight of block B
The weight of bracket A
The coefficient of static friction between block B and bracket A
The coefficient of kinetic friction between block B and bracket A
Calculation:
Let as consider the horizontal coordinate of A is
left of A.
Let as consider the horizontal coordinate of B is
left of A.
Let as consider
Sketch the system with coordinates points as shown in Figure (1).
Write the general equation of mass (m):
Here, W is the weight, g is the acceleration due to gravity.
Consider the constraint of cord.
Write the total length of cable length (L).
Here,
Differentiate Equation (1) with respect to t to write velocity of the blocks.
Here,
Differentiate Equation (2) with respect to t to write acceleration of the blocks.
Here,
No slip between bracket A and block B. Therefore, the acceleration of bracket A is equal to acceleration of block B.
Substitute
From Equation (4) and (5), consider
Sketch the free body diagram and kinetic diagram of bracket A as shown in Figure (2).
Refer Figure (2).
Apply Newton’s law of equation along x-axis.
Here, T is the tension in the cable,
Substitute
Here,
Sketch the free body diagram and kinetic diagram of block B as shown in Figure (3).
Refer Figure (3).
Apply Newton’s law of equation along x-axis.
Substitute
Here,
Apply Newton’s law of equation along y-axis.
Here,
Write the equation of frictional force
Substitute
Sketch the free body diagram and kinetic diagram of block C as shown in Figure (4).
Refer Figure (4).
Apply Newton’s law of equation along y-axis.
Substitute
Here,
Adding Equation (5), (6), and (9).
Substitute Equation (6) in Equation (12).
Subtract Equation (6) and (9).
Substitute Equation (6) and (10) in Equation (14).
Subtracting Equation (13) and (9).
Substitute 0.30 for
Thus, the maximum weight of block C if block B is not to slide on bracket A is
(b)
Find the accelerations of bracket A, block B and block C.
Answer to Problem 12.31P
The accelerations of bracket A is
The accelerations of block B is
The accelerations of block C is
Explanation of Solution
Given information:
The weight of block C is 10 % larger than the answer found in part (a).
Calculation:
Find the weight of block C.
The slip is occurring. Therefore consider the kinetic acceleration.
Write the equation of frictional force
Substitute
Subtract Equation (6) and (9).
Substitute Equation (17) in Equation (18).
Substitute 0.25 for
Rewrite the Equation (5).
Substitute 10 lb for
Find the accelerations of bracket A, block B and block C.
Solve Equation (3), (20), and (21).
Thus, the accelerations of bracket A is
Thus, the accelerations of block B is
Thus, the accelerations of block C is
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