Question

If 3 point masses 1, 2, & 3 kg are rigidly fixed on a massless triangular frame and placed on smooth frictionless table as shown below then choose correct statement base on given information.

• A. Acceleration of 3 kg must be $3\hat{j}$ m/s².
• B. Acceleration of 2 kg must be $-1\hat{j}$ m/s².
• C. Acceleration of 1 kg must be $6\hat{i}$ m/s².
• D. Acceleration of COM must be $0.5\hat{i} + 1.5\hat{j}$ m/s².

Answer 1

Answer: (D)

D

Answer 2

The problem describes a system of three point masses rigidly fixed on a massless triangular frame, placed on a smooth frictionless table. External forces are applied to each mass. We need to determine the correct statement regarding their accelerations.

1. Identify the system and applied forces:

• Masses: $m_1 = 1$ kg, $m_2 = 2$ kg, $m_3 = 3$ kg.
• Total mass of the system: $M_{total} = m_1 + m_2 + m_3 = 1 + 2 + 3 = 6$ kg.
• Applied forces:
  • On 1 kg mass: $\vec{F}_1 = 6\hat{i}$ N
  • On 2 kg mass: $\vec{F}_2 = -3\hat{i}$ N
  • On 3 kg mass: $\vec{F}_3 = 9\hat{j}$ N

2. Calculate the net external force on the system: The net external force is the vector sum of all forces applied to the masses. $\vec{F}_{net, ext} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3$ $\vec{F}_{net, ext} = (6\hat{i}) + (-3\hat{i}) + (9\hat{j})$ $\vec{F}_{net, ext} = (6 - 3)\hat{i} + 9\hat{j}$ $\vec{F}_{net, ext} = 3\hat{i} + 9\hat{j}$ N

3. Calculate the acceleration of the center of mass (COM): According to Newton's Second Law for a system of particles, the net external force equals the total mass times the acceleration of the center of mass: $\vec{F}_{net, ext} = M_{total} \vec{a}_{COM}$ $\vec{a}_{COM} = \frac{\vec{F}_{net, ext}}{M_{total}}$ $\vec{a}_{COM} = \frac{3\hat{i} + 9\hat{j}}{6}$ $\vec{a}_{COM} = \frac{3}{6}\hat{i} + \frac{9}{6}\hat{j}$ $\vec{a}_{COM} = 0.5\hat{i} + 1.5\hat{j}$ m/s²

4. Evaluate the given options:

• (A) Acceleration of 3 kg must be $3\hat{j}$ m/s².

  For a rigid body, the acceleration of an individual mass is generally not simply $\vec{F}_{applied}/m$. The internal forces from the rigid frame also act on each mass. If this were true, the net force on 3 kg mass would be $3 \text{ kg} \times 3\hat{j} \text{ m/s}^2 = 9\hat{j}$ N. This would imply the internal force from the frame on the 3 kg mass is zero, which is unlikely for a rigid body undergoing general motion (translation and rotation).

• (B) Acceleration of 2 kg must be $-1\hat{j}$ m/s².

  If this were true, the net force on 2 kg mass would be $2 \text{ kg} \times (-1\hat{j}) \text{ m/s}^2 = -2\hat{j}$ N. The applied force on 2 kg mass is $-3\hat{i}$ N. This would require a significant internal force in the y-direction, which is not directly supported without more information.

• (C) Acceleration of 1 kg must be $6\hat{i}$ m/s².

  If this were true, the net force on 1 kg mass would be $1 \text{ kg} \times 6\hat{i} \text{ m/s}^2 = 6\hat{i}$ N. This would imply the internal force from the frame on the 1 kg mass is zero, which is unlikely for a rigid body undergoing general motion.

• (D) Acceleration of COM must be $0.5\hat{i} + 1.5\hat{j}$ m/s².

  This matches our calculated value for the acceleration of the center of mass. This statement is always true for any system of particles, regardless of whether it's a rigid body or undergoing rotation, as long as the net external force and total mass are correctly identified.

5. Consideration of rigid body motion: For a rigid body, the accelerations of individual particles are generally different due to rotational motion, unless the body undergoes pure translation. For pure translation, the net torque about the center of mass must be zero. Without the exact coordinates of the masses, we cannot calculate the net torque and thus cannot definitively say if there is rotational motion. However, even if there is rotational motion, the calculation for the acceleration of the center of mass remains valid. The options (A), (B), and (C) make very specific claims about individual accelerations that are generally not true for a rigid body unless specific conditions (like zero internal forces or pure translation) are met, which are not guaranteed by the problem statement. Therefore, option (D) is the only statement that is fundamentally correct and derivable from the given information.