Blocks A and B, of masses ma and mb are at rest on a frictionless surface as shown in the figure, with block A fixed to the table. Block C of mass mc is suspended by a string that is tied to block B over an ideal pulley. Which of the following gives the magnitude of the force exerted by block A on block B?
A. (mb)g
B. (mc)g
C. (ma•mc)g / (ma+mb)
D. (mb•mc)g / (ma+mb)
The Correct Answer and Explanation is :
The correct answer is D. ((m_b \cdot m_c)g / (m_a + m_b)).
Explanation:
To analyze this system, let’s break it down step by step:
- Block A is fixed to the table, so it doesn’t move, and it only applies a force on block B.
- Block B, which is in contact with block A, can move, and it’s connected to block C via an ideal pulley and a string.
- Block C is suspended and has mass (m_c), and it exerts a force due to its weight, (m_c g), pulling downwards.
- Block B will experience two forces: the gravitational force acting downwards (which is (m_b g)) and the tension in the string due to block C, which is pulling upwards. Block A must exert a force on block B to counteract these forces.
Step 1: Understanding the forces acting on block C and block B
Block C is hanging and thus exerts a downward force on the string, which creates a tension force (T) in the string. This tension is the same throughout the string due to the ideal pulley, meaning the tension on block B is equal to the tension on block C.
The weight of block C is given by:
[
F_{\text{gravity, C}} = m_c g
]
This force causes a tension (T) in the string, which will act upward on block B.
Step 2: Analyzing block B
Block B will be pulled by the tension in the string upward, but block A will exert a horizontal force to counteract the forces acting on block B. The total force exerted by block A on block B is related to the combined effect of the masses of block B and block C.
To maintain equilibrium (since block A is fixed), the force exerted by block A on block B must balance out the effect of both blocks, leading to the expression:
[
F_{\text{A on B}} = \frac{(m_b \cdot m_c)g}{m_a + m_b}
]
This equation accounts for both the mass of block B (which influences its resistance to motion) and the mass of block C (which influences the pulling force due to gravity).
Thus, D is the correct answer.