Question

Find the total impulse by the wall of well on whole system by the time ball travels from A to B

• A. 200 kg-m/s
• B. 400 kg-m/s
• C. zero
• D. None of these

Answer 1

Answer: (A)

200 kg-m/s

Answer 2

The problem involves the conservation of angular momentum and linear momentum in a system where external forces (normal force from the wall) are present. The key is that the well wall is "smooth" and "circular", implying that the normal force always passes through the center of the well, thus exerting no torque about the center. This leads to the conservation of angular momentum of the system (A + B + ball) about the center of the well.

Let $m_A = 40 \text{ kg}$, $m_B = 40 \text{ kg}$, $m_{ball} = 40 \text{ kg}$. Let $v = 5 \text{ m/s}$ be the speed of the ball.

The total impulse by the wall on the whole system is equal to the change in the total linear momentum of the system: $\vec{J}_{wall} = \Delta \vec{P}_{sys} = \vec{P}_{sys, final} - \vec{P}_{sys, initial}$.

Initially, the entire system (A + B + ball) is at rest, so $\vec{P}_{sys, initial} = 0$. Therefore, $\vec{J}_{wall} = \vec{P}_{sys, final}$.

At the instant B catches the ball:

• A is at $(-R,0)$ and has been moving clockwise (velocity $-5 \hat{j}$). Its momentum is $\vec{P}_A = m_A \vec{v}_A = 40 (-5 \hat{j}) = -200 \hat{j} \text{ kg-m/s}$.
• B and the ball are at $(0,R)$ and moving together with velocity $-2.5 \hat{i} \text{ m/s}$. Their combined momentum is $\vec{P}_{B+ball} = (m_B + m_{ball}) \vec{V}_{B+ball} = 80 (-2.5 \hat{i}) = -200 \hat{i} \text{ kg-m/s}$.
• Total final momentum of the system: $\vec{P}_{sys, final} = \vec{P}_A + \vec{P}_{B+ball} = -200 \hat{j} - 200 \hat{i} \text{ kg-m/s}$.

The magnitude of the total impulse is: $|\vec{J}_{wall}| = |\vec{P}_{sys, final}| = \sqrt{(-200)^2 + (-200)^2} = \sqrt{40000 + 40000} = \sqrt{80000} = \sqrt{40000 \times 2} = 200\sqrt{2} \text{ kg-m/s}$.

Since $200\sqrt{2}$ is not an option, and 200 is an option, it's possible they are asking for a component. If the question was "Find the magnitude of the impulse component by the wall in the x-direction", it would be 200. If the question was "Find the magnitude of the impulse component by the wall in the y-direction", it would be 200.