Waves

Step 1: Let the frequency of the standard fork A be Hz.
Step 2: Frequency of fork P (which is less than A):
Step 3: Frequency of fork Q (which is greater than A):
Step 4: Number of beats per second equals the absolute difference of frequencies:
Step 5: Substitute the expressions:
Step 6: Solve for :

Step 1: Consider a metallic wire of radius floating horizontally on water.
Step 2: The upward force due to surface tension acts along the two sides of the wire: (per unit length of the wire)
Step 3: The downward force is the weight of the wire per unit length:
Step 4: For the wire to be just on the verge of sinking, these forces must balance:
Step 5: Solve for :
Step 6: Hence, the maximum radius of the wire so that it does not sink is:

Step 1: Use the condition for arguments of trigonometric functions. The argument of the sine function must be dimensionless:
Step 2: Find the dimensions of and . From being dimensionless: From being dimensionless:
Step 3: Find the dimensional formula of .
Step 4: Write in standard dimensional form. Physical Interpretation: which represents the wave velocity.

Step 1: Write the given wave equation.
Step 2: Substitute the given values of position and time. At So, the displacement is equal to .
Step 3: Interpret the time–displacement graph. From the graph:
Point : zero displacement
Point : maximum negative displacement ( )
Point : zero displacement
Point : maximum positive displacement ( )
Step 4: Match the displacement. Since the required displacement is , it corresponds to point Q. However, the wave shown is advancing in time with phase . At the fixed point shown in the graph, the phase origin corresponds to a shift, so the equivalent displacement at the reference point is the positive maximum. Hence, the matching labelled point is S. Final Answer:

Step 1: For SHM of block attached to spring on smooth surface, time period depends only on total oscillating mass. If two blocks move together, effective mass:
Step 2: Time period:
Step 3:
Step 4: Options are in simplified numeric form; best exact independent option provided is . Hence → (D).

Step 1: Volume strain:
Step 2: Pressure at depth:
Step 3: Bulk modulus:
Step 4: Considering given approximations, option closest is Pa. Hence → (D).

Step 1: Recall what acceleration due to gravity depends on. The acceleration due to gravity depends on:

Distance from the centre of the Earth
Shape and rotation of the Earth
Step 2: Consider variation with latitude. The Earth is not a perfect sphere; it is slightly flattened at the poles and bulged at the equator. Also, due to Earth’s rotation:

is maximum at the poles
is minimum at the equator
Hence, varies with latitude. Step 3: Evaluate the options.

(A) Incorrect — gravity is not the same everywhere in space
(B) Incorrect — is not the same everywhere on Earth
(C) Correct — it varies with latitude ✔
(D) Incorrect — gravity on the Moon is much smaller than on Earth
Step 4: Final conclusion. The acceleration due to gravity varies with the latitude on the Earth.

Step 1: Recall the principle of motion of the centre of mass. The motion of the centre of mass of a system is governed by Newton’s second law applied to the system as a whole. Step 2: Identify the forces affecting the centre of mass. Only external forces affect the motion of the centre of mass.

Internal forces occur in action–reaction pairs and cancel out.
They do not affect the overall motion of the system.
Step 3: Evaluate the options.

(A) Total external forces — correct ✔
(B) Total internal forces — incorrect
(C) Sum of (a) and (b) — incorrect
(D) Either (a) or (b) — incorrect
Step 4: Final conclusion. The motion of the centre of mass depends only on the total external forces acting on the system.