Mechanics

An object of mass 2000 g possesses 100 J kinetic energy. The object must be moving with a speed of

[amp_mcq option1=”10.0 m/s” option2=”11.1 m/s” option3=”11.2 m/s” option4=”12.1 m/s” correct=”option1″]

The correct option is A. The kinetic energy (KE) of an object is given by the formula $\text{KE} = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the speed. Given $\text{KE} = 100 \, \text{J}$ and $m = 2000 \, \text{g} = 2 \, \text{kg}$, we can solve for $v$.

Substitute the given values into the kinetic energy formula: $100 \, \text{J} = \frac{1}{2} \times (2 \, \text{kg}) \times v^2$. This simplifies to $100 = 1 \times v^2$, so $v^2 = 100$. Taking the square root of both sides gives $v = 10 \, \text{m/s}$ (since speed is a magnitude and must be non-negative).

It is important to ensure that all units are in the standard SI system before calculation. Mass is given in grams and must be converted to kilograms (1 kg = 1000 g). Energy is given in Joules, which is the standard SI unit.

A uniform motion of a car along a circular path experiences

[amp_mcq option1=”a change in speed due to a change in its direction of motion.” option2=”a change in velocity due to a change in its direction of motion.” option3=”a change in momentum due to no change in its direction of motion.” option4=”a constant momentum due to a change in its direction of motion.” correct=”option2″]

Uniform motion along a circular path means the object moves at a constant speed. However, velocity is a vector quantity, defined by both magnitude (speed) and direction. In circular motion, the direction of motion is constantly changing as the object moves along the curve. Since the direction of motion is changing, the velocity of the car is also continuously changing, even though its speed is constant.

Speed is the magnitude of velocity. In uniform circular motion, speed is constant. Velocity is a vector (speed + direction). The direction of motion is always tangent to the circular path and continuously changes. Therefore, velocity changes. Momentum is mass times velocity (p = mv). Since velocity changes, momentum also changes.

Since the velocity is changing, there is acceleration. This acceleration is called centripetal acceleration, and it is directed towards the center of the circle. It is responsible for changing the direction of the velocity vector. Uniform circular motion is an example of accelerated motion even though the speed is constant.

Work is said to be one Joule when a force of

[amp_mcq option1=”4 N moves an object by 25 cm.” option2=”2 N moves an object by 1 m.” option3=”1 N moves an object by 1 cm.” option4=”1 N moves an object by 50 cm.” correct=”option1″]

Work done by a force is defined as the product of the force applied and the displacement of the object in the direction of the force. The SI unit of work is the Joule (J). One Joule of work is done when a force of one Newton (N) moves an object by one meter (m) in the direction of the force.

The formula for work (W) is W = F × d, where F is the force and d is the displacement in the direction of the force. 1 Joule = 1 Newton × 1 meter. We need to check which option results in W = 1 J.
Option A: F = 4 N, d = 25 cm = 0.25 m. W = 4 N × 0.25 m = 1 N·m = 1 J.
Option B: F = 2 N, d = 1 m. W = 2 N × 1 m = 2 J.
Option C: F = 1 N, d = 1 cm = 0.01 m. W = 1 N × 0.01 m = 0.01 J.
Option D: F = 1 N, d = 50 cm = 0.5 m. W = 1 N × 0.5 m = 0.5 J.

Work is a scalar quantity. If the force and displacement are not in the same direction, the work done is calculated as W = Fd cos(θ), where θ is the angle between the force and displacement vectors. In the context of this question, it is implied that the force and displacement are in the same direction.

Fundamental laws of physics require

[amp_mcq option1=”conservation of energy and non-conservation of charge.” option2=”conservation of charge and non-conservation of linear momentum.” option3=”conservation of charge and non-conservation of energy.” option4=”conservation of energy, momentum and charge.” correct=”option4″]

Fundamental laws of physics are based on core principles, including several conservation laws. Energy, linear momentum, angular momentum, and electric charge are quantities that are conserved in isolated systems according to fundamental physical principles.

Conservation means that the total amount of the quantity remains constant over time within a closed system, even though it may be transformed or transferred between different forms or parts of the system. These conservation laws are derived from symmetries in nature and are cornerstones of physics.

Conservation of energy (first law of thermodynamics) states that energy cannot be created or destroyed, only transformed. Conservation of momentum (linear and angular) arises from Newton’s laws and implies that the total momentum of a system remains constant in the absence of external forces or torques. Conservation of charge states that the net electric charge of an isolated system remains constant.

Weight and mass of an object are defined with Newton’s laws of motion. Which among the following is true ?

[amp_mcq option1=”Weight is a constant of proportionality.” option2=”Mass is a constant of proportionality.” option3=”Mass is not a constant of proportionality.” option4=”Weight is a universal constant.” correct=”option2″]

Newton’s second law of motion states that the acceleration (a) of an object is directly proportional to the net force (F) acting on it and inversely proportional to its mass (m). Mathematically, F = ma. In this equation, mass (m) is the constant of proportionality relating the force to the acceleration it produces.

Mass is an intrinsic property of an object that measures its inertia (resistance to change in motion) and its gravitational pull. Weight (W) is the force exerted on an object due to gravity (W = mg), where g is the acceleration due to gravity.

Mass is a fundamental constant of proportionality in Newton’s second law and is a universal property of the object. Weight is a force and is not a universal constant; it varies depending on the strength of the gravitational field (value of g).

Buoyancy is a/an

[amp_mcq option1=”upward pressure” option2=”downward pressure” option3=”downward force” option4=”upward force” correct=”option4″]

Buoyancy is the upward force exerted by a fluid (liquid or gas) on an object immersed in it. This force opposes the weight of the object, which acts downwards.

According to Archimedes’ principle, the buoyant force is equal to the weight of the fluid displaced by the object. It is caused by the pressure difference between the bottom and top surfaces of the object due to the weight of the fluid column above them.

Pressure is force per unit area. While buoyancy arises from pressure differences, buoyancy itself is defined as a force. Forces have direction, and the buoyant force always acts upwards.

A liquid is kept in a glass beaker. Which one of the following statements is correct regarding the pressure exerted by the liquid column at the base of the beaker ?

[amp_mcq option1=”The pressure depends on the area of the base of the beaker” option2=”The pressure depends on the height of liquid column” option3=”The pressure does not depend on the density of the liquid” option4=”The pressure neither depends on the area of the base of the beaker nor on the height of liquid column” correct=”option2″]

The pressure exerted by a liquid column at the base of the beaker depends on the height of the liquid column.

– The pressure exerted by a column of liquid at a depth ‘h’ is given by the formula P = ρgh, where:
– P is the pressure
– ρ (rho) is the density of the liquid
– g is the acceleration due to gravity
– h is the height of the liquid column

– The formula P = ρgh shows that liquid pressure depends directly on the density of the liquid, the acceleration due to gravity, and the height (or depth) of the liquid column.
– It is independent of the shape of the container or the area of the base, as long as the height and density are the same (this is known as the hydrostatic paradox).

Two bodies of mass M each are placed R distance apart. In another system, two bodies of mass 2M each are placed $\frac{R}{2}$ distance apart. If F be the gravitational force between the bodies in the first system, then the gravitational force between the bodies in the second system will be

[amp_mcq option1=”16 F” option2=”1 F” option3=”4 F” option4=”None of the above” correct=”option1″]

The correct answer is A) 16 F.

The gravitational force between two bodies of masses $m_1$ and $m_2$ separated by a distance R is given by Newton’s Law of Gravitation: $F = G \frac{m_1 m_2}{R^2}$, where G is the gravitational constant.
In the first system: $m_1 = M$, $m_2 = M$, $R_1 = R$. The force is $F_1 = G \frac{M \times M}{R^2} = G \frac{M^2}{R^2}$. This force is given as F. So, $F = G \frac{M^2}{R^2}$.
In the second system: $m_1′ = 2M$, $m_2′ = 2M$, $R_2 = R/2$. The force is $F_2 = G \frac{(2M) \times (2M)}{(R/2)^2}$.
Calculate $F_2$: $F_2 = G \frac{4M^2}{R^2/4} = G \frac{4M^2}{R^2} \times 4 = 16 G \frac{M^2}{R^2}$.
Substitute the expression for F: $F_2 = 16 F$.

The gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers. Doubling the masses quadruples the product of masses ($2M \times 2M = 4M^2$). Halving the distance quarters the squared distance ($(R/2)^2 = R^2/4$), meaning the force is multiplied by 4 due to the inverse square law ($1 / (1/4) = 4$). The combined effect is a multiplication by $4 \times 4 = 16$.

A pendulum clock is lifted to a height where the gravitational acceleration has a certain value g. Another pendulum clock of same length but of double the mass of the bob is lifted to another height where the gravitational acceleration is g/2. The time period of the second pendulum would be :
(in terms of period T of the first pendulum)

[amp_mcq option1=”$\sqrt{2}$ T” option2=”$\frac{1}{\sqrt{2}}$ T” option3=”$2\sqrt{2}$ T” option4=”T” correct=”option1″]

The correct answer is A) $\sqrt{2}$ T.

The time period (T) of a simple pendulum is given by the formula $T = 2\pi \sqrt{\frac{L}{g}}$, where L is the length of the pendulum and g is the acceleration due to gravity. The mass of the bob does not affect the time period of a simple pendulum.
For the first pendulum: Length $L_1=L$, gravity $g_1=g$. Time period $T_1 = 2\pi \sqrt{\frac{L}{g}} = T$.
For the second pendulum: Length $L_2=L$ (stated as same length), mass $M_2=2M_1$ (mass does not affect T), gravity $g_2=g/2$. Time period $T_2 = 2\pi \sqrt{\frac{L_2}{g_2}} = 2\pi \sqrt{\frac{L}{g/2}} = 2\pi \sqrt{\frac{2L}{g}}$.
We can rewrite $T_2$ in terms of $T_1$: $T_2 = \sqrt{2} \times (2\pi \sqrt{\frac{L}{g}}) = \sqrt{2} T_1 = \sqrt{2} T$.

The time period of a simple pendulum is independent of the mass and amplitude (for small oscillations) of the bob. It depends only on the length of the string and the local acceleration due to gravity. The question tests the understanding that mass does not influence the time period and how the time period scales with changes in gravitational acceleration.

A car starts from Bengaluru, goes 50 km in a straight line towards south, immediately turns around and returns to Bengaluru. The time taken for this round trip is 2 hours. The magnitude of the average velocity of the car for this round trip

[amp_mcq option1=”is 0.” option2=”is 50 km/hr.” option3=”is 25 km/hr.” option4=”cannot be calculated without knowing acceleration.” correct=”option1″]

Average velocity is defined as the total displacement divided by the total time taken. Displacement is the change in position from the starting point to the ending point. The car starts from Bengaluru and returns to Bengaluru. Therefore, the initial position and the final position are the same. This means the total displacement is zero.
Total displacement = Final position – Initial position = Bengaluru – Bengaluru = 0 km.
Total time taken = 2 hours.
Average velocity = Total displacement / Total time = 0 km / 2 hours = 0 km/hr.
The magnitude of the average velocity is the absolute value of the average velocity, which is $|0| = 0$.

– Average velocity is a vector quantity and depends on displacement, not total distance traveled.
– Displacement is the shortest straight-line distance from the starting point to the ending point, along with direction. If the start and end points are the same, displacement is zero.
– Average speed is a scalar quantity and depends on the total distance traveled divided by the total time. In this case, the average speed is 100 km / 2 hr = 50 km/hr.

– The path taken (straight line towards south and back) is relevant for calculating distance but not displacement when the trip is a round trip back to the origin.
– The acceleration is not constant during the trip (it changes direction when turning around), but this information is not needed to calculate average velocity, which only depends on total displacement and total time.