Concept explainers
Consider the three connected objects shown in Figure P5.88. Assume first that the inclined plane is friction-less and that the system is in equilibrium. In terms of m, g, and θ, find (a) the mass M and (b) the tensions T, and T2. Now assume that the value of Af is double the value found in part (a). Find (c) the acceleration of each object and (d) the tensions T1 and T2. Next, assume that the coefficient of static friction between m and 2m and the inclined plane is m, and that the system is in equilibrium. Find (e) the maximum value of M and (0 the minimum value of M. (g) Compare the values of T2 when M has its minimum and maximum values.
(a)
The expression for the mass of
Answer to Problem 5.88AP
The expression for the mass
Explanation of Solution
The free body diagram of the three connected objects is shown in Figure below,
Figure (1)
From Figure (1), the equilibrium forces acts on the object of mass
Here,
From the Newton’s second law of motion, the net force on the object of mass
Here,
Substitute
From Figure (1), the equilibrium forces act on the object of mass
Here,
From the Newton’s second law of motion, the net force on the object of mass
Substitute
Add the equation (1) with equation (2).
From Figure (1), the equilibrium forces act on the object of mass
Here,
Substitute
From the Newton’s second law of motion, the net force on the object of mass
Substitute
The system is in equilibrium so value of the acceleration is zero.
Substitute
Conclusion:
Therefore, the expression for the mass
(b)
The expressions for tensions
Answer to Problem 5.88AP
The expression for the tension
Explanation of Solution
From part (a), the expression for the mass of
From part (a), the equilibrium forces act on the object is,
The system is in equilibrium, the value of acceleration is zero so the net force acts on the system is also zero.
Substitute
Substitute
Thus, the expression for the tension
From part (a), the equation (2) is,
Substitute
Substitute
Thus, the expression for the tension
Conclusion:
Therefore, the expression for the tension
(c)
The acceleration of each object.
Answer to Problem 5.88AP
The acceleration of each object is
Explanation of Solution
Given info: The value of mass
From part (a), the expression for the mass when it is double represents as,
From part (a), the equation (1) is,
Rearrange the above equation.
From part (a), the equation (2) is,
Rearrange the above equation.
Substitute
From Figure (1), the equilibrium forces act on the object of mass
Substitute
Subtract the equation (3) from equation (4).
Conclusion:
Therefore, the acceleration of each object is
(d)
The expressions for tensions
Answer to Problem 5.88AP
The expression for the tension
Explanation of Solution
From part (c), the expression for the acceleration is,
From part (c), the equation for tension
Substitute
Thus, the expression for tension
From part (c), the equation (3) is,
Substitute
Thus, the expression for tension
Conclusion:
Therefore, the expression for the tension
(e)
The maximum value of
Answer to Problem 5.88AP
The maximum value of
Explanation of Solution
Given info: The coefficient of static friction between mass
The static friction forces on the masses
Figure (2)
From the Figure (2), the normal force on the object of mass
The expression for the static friction force on the object of mass
Here,
Substitute
From the Figure (1), the equilibrium forces acts on the object of mass
Substitute
Rearrange the above equation for
Substitute
From the Figure (2), the normal force on the object of mass
Substitute
From the Figure (1), the equilibrium forces acts on the object of mass
Substitute
Add equation (4) with the above equation.
Substitute
The equilibrium forces on the block of mass
Substitute
Conclusion:
Therefore, the maximum value of
(f)
The minimum value of
Answer to Problem 5.88AP
The minimum value of
Explanation of Solution
From part (e), the expression of tension
The equilibrium forces acts on block for minimum mass of
Substitute
Conclusion:
Therefore, the minimum value of
(g)
The difference between the tension
Answer to Problem 5.88AP
The difference between the tension for maximum and minimum mass is
Explanation of Solution
From part (e), the expression for maximum mass is,
From part (f), the expression for the minimum mass is,
From part (e), the expression for the tension for maximum mass is,
From part (f), the expression for the tension for maximum mass is,
Compare both the above equation.
Substitute
Conclusion:
Therefore, the difference between the tension for maximum and minimum mass is
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