Work-energy-power practice · Mixed difficulty

Option A ($\frac{M_2}{M_1}$) is correct because it is derived from the conservation of momentum and the definition of kinetic energy.

The correct answer is A. Statement I is true because the overturning tendency is related to the centripetal force, which increases with the square of the velocity. Statement II correctly explains this relationship with the formula for the banking angle.

Option A (4.9 J) is correct because it is the loss of kinetic energy calculated after the bullet gets embedded in the sandbag using conservation of momentum.

Option B (62.5 cm) is correct because after each collision, the height is reduced by a factor of $e^2$, where e is the coefficient of restitution.

Option C is correct because the ratio of linear momenta is the square root of the ratio of the masses when kinetic energies are equal.

Option A (Potential energy at point $x$ when $U=0$ at $x=0$ is $\frac{d x^4}{4}$) is correct because the potential energy is the integral of the force with respect to displacement.

Option D (4 sec) is correct because it is the time period calculated using the condition that the water just does not fall down, implying the minimum speed at the top is $\sqrt{gr}$.

Option D ($\frac{{mv_1^2t}}{{t_1^2}}$) is correct because the instantaneous power is calculated using $P = Fv$, where $F = ma$ and $a = \frac{{v_1}}{{t_1}}$, leading to the given expression.

Statement-I & II are true. The work done by all forces equals the change in kinetic energy, and internal forces can do positive work.

Option B (125) is correct because the angular velocity is calculated using the formula $\omega = v/r$, where v is the linear velocity and r is the radius.

Option A (If both assertion and reason are true and reason is the correct explanation of the assertion) is correct because both the cyclist leaning inwards and the man leaning outwards are due to the need to provide the necessary centripetal force, and the reason correctly states the direction of centripetal acceleration.

Option D (20 J) is correct because the work done is calculated by multiplying the force component along the displacement with the displacement.

Option A (2) is correct because the ratio of linear speeds is calculated using $v = r\omega$, so $v_1/v_2 = r_1/r_2$ since $\omega$ is the same.

Option D (All of the above) is correct because frictional force can do positive, negative, or zero work depending on the frame of reference and the direction of motion.

Option A (57 J) is correct because the work done is the dot product of the force and the displacement vector.

The correct answer is A. During downward motion, the vertical component of velocity increases, and since power is the dot product of force and velocity, the power delivered by gravity increases.

Option D ($3/\sqrt 2$) is correct because the maximum speed occurs when the kinetic energy is maximum. Since $E = KE + PE$, $KE = E - PE$. We need to find the minimum potential energy and then calculate the maximum kinetic energy and corresponding speed.

The correct answer is A. When the bookcase tilts, its center of gravity rises, increasing its gravitational potential energy, which then causes it to return to its upright position.

Option A (Force) is correct because force is invariant under Galilean transformations, meaning it remains the same in different inertial frames.

The correct answer is D. In non-uniform accelerated circular motion, the velocity is always tangential (transverse), and the acceleration has both radial (centripetal) and tangential components because both the direction and magnitude of the velocity are changing.

Option D ($Fv\;$) is correct because power is the product of force and velocity when the force is constant and the velocity is constant.

Option C ($\frac{{2u}}{{1 + \frac{M}{m}}}$) is correct because it is derived from conservation of momentum and kinetic energy in an elastic collision.

Option D (The liner momentum is conserved) is correct because, in any collision (elastic or inelastic), the total linear momentum of the system is always conserved in the absence of external forces.

Option A (The light body) is correct because for equal momenta, kinetic energy is inversely proportional to mass; hence, the lighter body has greater kinetic energy.

Option C is correct because only the y-component of the force contributes to the work done when the displacement is along the y-axis.

Option A is correct because the force is always perpendicular to the displacement, hence no work is done.

Option D ($\frac{{MgL}}{{18}}$) is correct because the work done is equal to the change in potential energy of the hanging part of the chain.

Option A (Zero) is correct because the wall's displacement is zero, hence no work is done on the wall.

Option B ($\frac{{ - 2Mgd}}{3}$) is correct because the work done by the rope is negative as it acts upwards while the block moves downwards, and the tension is less than the weight due to the downward acceleration.

The correct answer is B. For a particle moving in a circle with constant angular speed, the acceleration vector points towards the center of the circle (centripetal acceleration), and the velocity vector is tangent to the circle. Therefore, the statement that the acceleration vector is tangent to the circle is false.