Mechanics

Step 1: Understanding the Concept:
For a point body (or a single point on a rigid body) to be in equilibrium under the action of forces, the net force (vector sum of all forces) must be zero. This is Newton's First Law. We need to find the conditions for two forces, and , such that their vector sum is zero.
Step 2: Key Formula or Approach:
The condition for equilibrium is: This vector equation implies conditions on the magnitude and direction of the two forces.
Step 3: Detailed Explanation:
From the equilibrium condition , we can write . Let's analyze what this relationship means:
Magnitude: The magnitude of a vector is always non-negative. Taking the magnitude of both sides: . This means the forces must be equal in magnitude. So, statement (A) is true. Statement (E) is false.
Direction: The negative sign indicates that the vector points in the exact opposite direction to the vector . So, they must have opposite directions. Statement (C) is true. Statement (B) is false.
Line of Action: For the vectors to be exactly opposite, they must lie along the same straight line. This is also known as being collinear. So, statement (D) is true.
Step 4: Final Answer:
For two forces to be in equilibrium, they must be equal in magnitude, opposite in direction, and act along the same straight line. Therefore, statements (A), (C), and (D) are all required.

Step 1: Understanding the Concept:
This question requires applying the law of parallelogram of forces to find the magnitude of the resultant force (R) for two forces P and Q acting at a point with an angle between them.
Step 2: Key Formula or Approach:
The general formula for the magnitude of the resultant is: We will apply this formula to the specific cases given in List-I.
Step 3: Detailed Explanation:

(A) P and Q are perpendicular forces: This means the angle between them is . Since , the formula becomes: So, (A) matches with (IV).
(B) P and Q are equal forces ( ) acting at an angle : Substitute Q=P into the general formula: Using the half-angle identity : So, (B) matches with (III).
(C) P and Q are acting in the same direction: This means the angle between them is . Since , the formula becomes: So, (C) matches with (II).
(D) P and Q are acting in the opposite direction: This means the angle between them is . Since , the formula becomes: So, (D) matches with (I).
Step 4: Final Answer:
The correct pairings are (A)-(IV), (B)-(III), (C)-(II), (D)-(I). This corresponds to option (D).

Step 1: Understanding the Concept:
When an object is thrown vertically upwards, its initial kinetic energy is converted into gravitational potential energy. At the maximum height, the velocity is momentarily zero. We can use the kinematic equation relating initial velocity, final velocity, acceleration, and displacement.
Step 2: Key Formula or Approach:
The relevant kinematic equation is: At the maximum height ( ), the final velocity . The acceleration is . The displacement is . Solving for the maximum height : This shows that the maximum height is proportional to the square of the initial velocity ( ).
Step 3: Detailed Explanation:
Let the initial velocities of the two stones be and . We are given the ratio of their velocities: Let the maximum heights they reach be and . Using the formula , we can find the ratio of their heights: Substitute the given ratio of velocities: Step 4: Final Answer:
The ratio of the maximum heights attained by the stones is 4:25.