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A block in the shape of a square having mass M is at rest on a horizontal surface. As shown in the figure, two blocks of mass m, and m₂ 2 are in contact by a string passing through a pulley. Assuming friction to be negligible then a constant horizontal force of magnitude F that must be applied to M so that m, and , do not move with respect to M:?
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A block in the shape of a square having mass M is at rest on a horizon...
Understanding the System
The system consists of a square block of mass M on a horizontal surface, and two blocks of mass m₁ and m₂ connected by a string over a pulley. The goal is to find the horizontal force F needed to keep the blocks m₁ and m₂ stationary relative to M.
Forces Acting on Each Block
- Block M:
- It experiences a horizontal force F applied to it.
- The tension T in the string pulls to the left due to the weights m₁ and m₂.
- Block m₁:
- The weight m₁g acts downward, and the normal force N₁ acts upward.
- Tension T pulls block m₁ horizontally to the left.
- Block m₂:
- Similar forces act on m₂ as it hangs vertically.
- The weight m₂g acts downward, and the tension T acts upward.
Setting Up Equations
To ensure blocks m₁ and m₂ do not move with respect to M, the acceleration of block M must equal the acceleration of the system formed by m₁ and m₂.
- The net force on block M is given by:
- F - T = Ma (1)
- For the hanging block m₂:
- m₂g - T = m₂a (2)
- For the block m₁ on the surface:
- T = m₁a (3)
Solving the Equations
From equation (3), we can express T in terms of a:
- T = m₁a
Substituting T into equation (1):
- F - m₁a = Ma
- Therefore, F = Ma + m₁a
From equation (2), we can express a in terms of g:
- a = (m₂g - T) / m₂
- Substitute T from (3) into this to find a.
Final Result
To ensure m₁ and m₂ remain stationary with respect to M, the force F must be calculated using the derived expressions, ensuring that all forces are balanced in the system.
Thus, a precise value for F can be determined based on the masses M, m₁, and m₂, ensuring equilibrium.
The system consists of a square block of mass M on a horizontal surface, and two blocks of mass m₁ and m₂ connected by a string over a pulley. The goal is to find the horizontal force F needed to keep the blocks m₁ and m₂ stationary relative to M.
Forces Acting on Each Block
- Block M:
- It experiences a horizontal force F applied to it.
- The tension T in the string pulls to the left due to the weights m₁ and m₂.
- Block m₁:
- The weight m₁g acts downward, and the normal force N₁ acts upward.
- Tension T pulls block m₁ horizontally to the left.
- Block m₂:
- Similar forces act on m₂ as it hangs vertically.
- The weight m₂g acts downward, and the tension T acts upward.
Setting Up Equations
To ensure blocks m₁ and m₂ do not move with respect to M, the acceleration of block M must equal the acceleration of the system formed by m₁ and m₂.
- The net force on block M is given by:
- F - T = Ma (1)
- For the hanging block m₂:
- m₂g - T = m₂a (2)
- For the block m₁ on the surface:
- T = m₁a (3)
Solving the Equations
From equation (3), we can express T in terms of a:
- T = m₁a
Substituting T into equation (1):
- F - m₁a = Ma
- Therefore, F = Ma + m₁a
From equation (2), we can express a in terms of g:
- a = (m₂g - T) / m₂
- Substitute T from (3) into this to find a.
Final Result
To ensure m₁ and m₂ remain stationary with respect to M, the force F must be calculated using the derived expressions, ensuring that all forces are balanced in the system.
Thus, a precise value for F can be determined based on the masses M, m₁, and m₂, ensuring equilibrium.
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A block in the shape of a square having mass M is at rest on a horizontal surface. As shown in the figure, two blocks of mass m, and m₂ 2 are in contact by a string passing through a pulley. Assuming friction to be negligible then a constant horizontal force of magnitude F that must be applied to M so that m, and , do not move with respect to M:?
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