Block a is moving with a certain acceleration along a frictionless horizontal surface. When a second block B is placed on top of block A, the acceleration of the combined block drops to 1/5 the original value. What is the ratio of the mass of A to the mass of B :
The force that does the accelerating presumably stays the same. The acceleration rate drops by the same ratio that the total mass rises. That ratio is 5.
(MB + MA)/MA = 5
MB = 4 MA
To solve this problem, we need to use the concept of Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
Let's denote the mass of block A as ma and the mass of block B as mb. When block B is placed on top of block A, the system's total mass becomes ma + mb.
Given that the acceleration of the combined block (A + B) drops to 1/5 the original value, we can set up the following equation:
(a_combined) = 1/5 * (a_A)
where a_combined is the acceleration of the combined block (A + B) and a_A is the acceleration of block A.
Now, let's consider the forces acting on the system. Since the surface is frictionless, the only force acting on the combined block is the force of gravity.
For block A, the force of gravity is given by the equation:
F_A = ma * g
where g is the acceleration due to gravity.
For the combined block (A + B), the force of gravity is given by:
F_combined = (ma + mb) * g
Since the force is directly proportional to the acceleration, we can set up the following equation:
F_A = m_A * a_A
F_combined = (m_A + m_B) * a_combined
Dividing these two equations, we get:
(m_A * a_A) / ((m_A + m_B) * a_combined) = 1
Rearranging the equation, we have:
m_A * a_A = (m_A + m_B) * a_combined
Now, substitute the value of a_A from the given information:
m_A * (1/5 * a_A) = (m_A + m_B) * a_combined
Simplifying this equation, we get:
m_A / 5 = (m_A + m_B)
Multiplying through by 5:
m_A = 5 * (m_A + m_B)
Expanding:
m_A = 5m_A + 5m_B
Rearranging and simplifying:
4m_A = 5m_B
Dividing by m_B:
4m_A / m_B = 5
Therefore, the ratio of the mass of block A to the mass of block B is 4:5.