Concept explainers
The range of the values of the mass m2 so that the two connected blocks resting on two inclined planes are in equilibrium.
Answer to Problem 83QAP
The mass m2 can have values between 3.51 kg and 52.64 kg for the system of masses to be in equilibrium.
Explanation of Solution
Given info:
Formula used:
From the diagram given, if the value of m2 falls below a certain value, it would slide up the plane and the block of mass m1 would slide down. This gives the value of the minimum mass of m2.
If the value of m2 increases beyond a certain value, the block would slide down the plane and the block of mass m1 would slide up.
By drawing the free body diagrams for each case, and applying the force equations for each case, the range of the values of m2 can be determined.
Explanation and Calculation:
Case 1:
Consider the case when the block 1slides down the plane and the block 2 slides up the plane.
The free body diagram for m1 is shown below:
The block rests on an incline which makes an angle
The weight
Resolve the weight w1 into components along the +x and −y directions as shown. The magnitudes of the components are given by,
Write the force equation along the +y direction.
Since the block is in equilibrium,
Therefore,
The force of friction is given by,
The system is in equilibrium along the x direction too. Therefore,
Using equations (1) and (2), in the above equation,
In a similar manner, construct a free body diagram for the block 2.
The block rests on an incline which makes an angle
The weight
Resolve the weight w2 into components along the −x and −y directions as shown. The magnitudes of the components are given by,
Write the force equation along the +y direction and apply the condition for equilibrium.
Therefore,
The force of friction is given by,
The system is in equilibrium along the x direction too. Therefore,
Using equations (4) and (6), in the above equation,
Add equations (3) and (7) and write an expression for m2.
Substitute the known values of the variables in the expression and calculate the value of m2.
The minimum value of m2 for which the system is in equilibrium is 3.51 kg.
Case 2:
Consider the case when the block 1slides up the plane and the block 2 slides down the plane.
The free body diagram for m1 is shown below:
Assume the +x direction up the incline and the +y direction perpendicular to the plane. The weight
The equations (1) and (2) are applied to this free body diagram too.
Write the condition for equilibrium along the x direction.
Use equations (1) and (2) in the expression.
In a similar manner, construct a free body diagram for the block 2.
. Assume a coordinate system which has the +x direction pointing down the plane and the +y direction perpendicular to the plane away from it. The weight
The equations (4), (5) and (6) are valid for this case too.
Write the equation for equilibrium along the x direction.
Using equations (4) and (6) in the expression,
Add equations (8) and (9) and write an expression for m2.
Substitute the known values of the variables in the expression.
The maximum value of m2 for the system to be in equilibrium is 52.64 kg.
Conclusion:
Thus, the mass m2 can have values between 3.51 kg and 52.64 kg for the system of masses to be in equilibrium.
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Chapter 5 Solutions
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