Time: 1 hr
30 questions
The expression $(\vec{A}\times\vec{B}) \cdot \vec{B}$ represents a **scalar triple product**. The vector $(\vec{A}\times\vec{B})$ is a new vector that is perpendicular to both $\vec{A}$ and $\vec{B}$.
The dot product of two vectors is zero if they are perpendicular. Since $(\vec{A}\times\vec{B})$ is perpendicular to $\vec{B}$ by definition of the cross product, their dot product is zero.
Therefore, $(\vec{A}\times\vec{B}) \cdot \vec{B} = 0$.
The horizontal range ($R$) of a projectile launched with initial speed $v_0$ and launch angle $\theta$ is given by the formula:
$$R = \frac{v_0^2 \sin(2\theta)}{g}$$
To maximize the range, the term $\sin(2\theta)$ must be at its maximum value, which is 1. This occurs when the angle $2\theta$ is $90^{\circ}$.
$$2\theta = 90^{\circ} \implies \theta = 45^{\circ}$$
For a point on a rotating rigid body, the **tangential acceleration** ($a_t$) is the component of acceleration that is tangent to the circular path of the point. It is directly proportional to both the angular acceleration ($\alpha$) of the body and the radius ($r$) of the point's path.
The relationship is given by the formula: $$a_t = \alpha r$$
This is the rotational equivalent of the linear equation $a = \frac{dv}{dt}$.
In the case of **pure rolling**, the point of contact between the wheel and the surface has zero velocity. This condition ensures that the linear speed ($v$) of the center of mass is directly related to the angular speed ($\omega$) and the radius ($R$) of the wheel.
The relationship is given by the formula: $$v = \omega R$$
This equation holds true only when there is no slipping.
The **Reynolds number (Re)** is a dimensionless quantity used to predict fluid flow patterns. It represents the ratio of inertial forces to viscous forces within a fluid.
For flow in a pipe, the transition between laminar (smooth, orderly flow) and turbulent (chaotic, disorderly flow) is generally defined by the following ranges:
Therefore, a Reynolds number less than 2000 indicates a typically laminar flow.
A **soap bubble** has two free surfaces, an inner one and an outer one, unlike a liquid drop which has only one. Both surfaces contribute to the excess pressure due to surface tension.
The excess pressure ($\Delta P$) is given by the formula: $$\Delta P = \frac{4\gamma}{R}$$
The excess pressure for a liquid drop, which has only one surface, is $\frac{2\gamma}{R}$.
**Bernoulli's principle** states that for an ideal fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.
The equation is: $$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
If the height ($h$) remains the same, the term $\rho gh$ is constant. For the total value to remain constant, if the velocity ($v$) increases, the static pressure ($P$) must decrease.
The relationship between the diffraction angle ($\theta$), wavelength ($\lambda$), and grating spacing ($d$) is given by the **diffraction grating equation**:
$$d \sin\theta = m\lambda$$
where $m$ is the order of the diffraction maximum. Rearranging for the diffraction angle, we get:
$$\sin\theta = \frac{m\lambda}{d}$$
If $\lambda$ increases while $m$ and $d$ are fixed, the value of $\sin\theta$ must increase. Since $\sin\theta$ is a monotonically increasing function for angles between $0^{\circ}$ and $90^{\circ}$, the diffraction angle $\theta$ also **increases**.
When two or more thin lenses are placed in contact, their powers add up directly. This is because the overall effect on light is a simple sum of the individual effects of each lens.
The total power ($P_{eq}$) of the combination is the sum of the individual powers ($P_1, P_2, ...$).
$$P_{eq} = P_1 + P_2$$
This is a fundamental principle of thin lens combinations.
When an object is placed at the **center of curvature (C)** of a concave mirror, the image is formed at the same location. This is a special case in spherical mirror optics.
The image is:
The force on a current-carrying wire in a uniform magnetic field is given by the formula:
$$\vec{F} = I(\vec{L} \times \vec{B})$$
The magnitude of this force is the product of the magnitudes of the vectors and the sine of the angle between them ($\theta$).
$$F = ILB \sin\theta$$
If the wire is perpendicular to the magnetic field, $\theta = 90^\circ$ and $\sin(90^\circ) = 1$, so the force is at its maximum magnitude, $F = ILB$.
An **inductor** stores energy in its magnetic field when a current flows through it. The amount of energy ($E$) stored is proportional to the inductance ($L$) and the square of the current ($I$).
The formula for the stored energy is: $$E = \frac{1}{2}LI^2$$
This is analogous to the kinetic energy formula for a mass, $E = \frac{1}{2}mv^2$, where inductance ($L$) is the electrical equivalent of inertia (mass) and current ($I$) is the electrical equivalent of velocity.
When capacitors are connected in **series**, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances.
$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}$$
To find $C_{eq}$, we can combine the fractions on the right side:
$$\frac{1}{C_{eq}} = \frac{C_2 + C_1}{C_1C_2}$$
Taking the reciprocal of both sides gives the equivalent capacitance:
$$C_{eq} = \frac{C_1C_2}{C_1+C_2}$$
The charging of a capacitor in an RC circuit is described by the equation:
$$Q(t) = Q_{max}(1 - e^{-t/\tau})$$
where $Q(t)$ is the charge at time $t$, $Q_{max}$ is the final charge, and $\tau=RC$ is the time constant.
After one time constant, $t=\tau$:
$$Q(\tau) = Q_{max}(1 - e^{-\tau/\tau}) = Q_{max}(1 - e^{-1})$$
Since $e^{-1} \approx 0.368$, the charge is:
$$Q(\tau) \approx Q_{max}(1 - 0.368) = 0.632 Q_{max}$$
This means the charge on the capacitor reaches approximately **63%** of its final value.
The change in wavelength ($\Delta\lambda$) in **Compton scattering** is given by the Compton shift formula:
$$\Delta\lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)$$
where $\theta$ is the scattering angle.
To maximize $\Delta\lambda$, the term $(1 - \cos\theta)$ must be at its maximum. The maximum value of this term occurs when $\cos\theta$ is at its minimum value, which is -1. This happens at an angle of $\theta = 180^{\circ}$.
At $\theta = 180^{\circ}$, the photon is scattered directly backward, and the change in wavelength is maximum.
The penetrating power of radiation depends on its charge and mass. Here's a quick comparison:
Therefore, **gamma rays** are the most penetrating.
The area of a triangle formed by two vectors is half the magnitude of their cross product. The cross product of $\vec{a}$ and $\vec{b}$ is:
$$\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & 1 \\ 1 & 1 & 0 \end{vmatrix} = \hat{i}(0-1) - \hat{j}(0-1) + \hat{k}(2-0) = -\hat{i} + \hat{j} + 2\hat{k}$$
The magnitude of this vector is:
$$|\vec{a}\times\vec{b}| = \sqrt{(-1)^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$$
The area of the triangle is half of this magnitude:
$$\text{Area} = \frac{1}{2}|\vec{a}\times\vec{b}| = \frac{\sqrt{6}}{2} \text{ unit}^2$$
The volume of a parallelepiped formed by three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is the absolute value of their **scalar triple product**, which can be calculated using a determinant.
$$\text{Volume} = |\vec{a} \cdot (\vec{b} \times \vec{c})| = \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 1 & 0 & 1 \end{vmatrix}$$
Expanding the determinant:
$$\text{Volume} = 1(1 \cdot 1 - 2 \cdot 0) - 2(0 \cdot 1 - 2 \cdot 1) + 3(0 \cdot 0 - 1 \cdot 1)$$
$$\text{Volume} = 1(1) - 2(-2) + 3(-1) = 1 + 4 - 3 = 2$$
The volume is 2. Therefore, option C is the correct answer.
This is a classic vector problem. The motion of the boat can be analyzed in two independent components: across the river and downstream.
First, calculate the time ($t$) it takes for the boat to cross the river. The time is determined by the boat's velocity perpendicular to the river flow and the river's width.
$$t = \frac{\text{width}}{\text{boat speed}} = \frac{120~m}{5~m/s} = 24~s$$
Next, calculate the downstream displacement. During the 24 seconds it takes to cross the river, the river's current carries the boat downstream. The downstream displacement ($x$) is the product of the river's speed and the time to cross.
$$x = (\text{river speed}) \times t = 3~m/s \times 24~s = 72~m$$
The boat will land 72 m downstream from the point directly opposite its starting point.
The position of an object in simple harmonic motion (SHM) is given by:
$$x(t) = A\cos(\omega t)$$
To find the velocity, we take the first derivative with respect to time:
$$v(t) = \frac{dx}{dt} = -A\omega\sin(\omega t)$$
To find the acceleration, we take the second derivative:
$$a(t) = \frac{dv}{dt} = -A\omega^2\cos(\omega t)$$
The magnitude of acceleration is $|a(t)| = A\omega^2|\cos(\omega t)|$. The maximum value of $|\cos(\omega t)|$ is 1. Therefore, the maximum magnitude of acceleration ($a_{max}$) is:
$$a_{max} = \omega^2 A$$
The period ($T$) of a **simple pendulum** is the time it takes for one complete oscillation. For small-angle oscillations, the motion is approximately simple harmonic motion.
The formula for the period of a simple pendulum of length $l$ in a gravitational field with acceleration $g$ is:
$$T = 2\pi\sqrt{\frac{l}{g}}$$
This formula shows that the period is independent of the mass of the bob and the amplitude of the oscillation (for small angles).
The cylinder that reaches the bottom first will have a greater final linear velocity and thus a greater final kinetic energy. The total kinetic energy is the sum of translational and rotational kinetic energy:
$$KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$$
Using the no-slip condition $v = \omega R$, we can write:
$$KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I(\frac{v}{R})^2 = \frac{1}{2}v^2(m + \frac{I}{R^2})$$
The moment of inertia ($I$) for a solid cylinder is $I_{solid} = \frac{1}{2}mR^2$. The moment of inertia for a thin hollow cylinder is $I_{hollow} = mR^2$. Since $I_{hollow} > I_{solid}$, the hollow cylinder requires more energy to rotate.
For a given change in potential energy, the object that puts more energy into rotation will have less energy left for translation, and thus a lower final linear velocity. Since the solid cylinder has a smaller moment of inertia, it has more energy for translation and will reach the bottom first.
The **torque ($\vec{\tau}$)** is a measure of the force that causes an object to rotate. It is a vector quantity defined by the cross product of the position vector ($\vec{r}$) and the force vector ($\vec{F}$).
$$\vec{\tau} = \vec{r} \times \vec{F}$$
The magnitude of the torque is given by the formula:
$$\tau = |\vec{r}||\vec{F}|\sin\theta = Fr\sin\theta$$
where $\theta$ is the angle between the position vector (lever arm) and the force vector. The torque is caused only by the component of the force that is perpendicular to the lever arm, which is $F\sin\theta$.
The **gravitational force** is a **conservative force**. A key property of a conservative force is that the work done by it on an object moving around any closed path is always zero.
This is because the work done by a conservative force only depends on the initial and final positions of the object, not the path taken. Since the object returns to its starting point, its initial and final positions are the same, and the net work done is zero.
A **monoatomic ideal gas** has only three degrees of freedom, which correspond to translational motion in the x, y, and z directions. Each degree of freedom contributes $\frac{1}{2}RT$ to the internal energy per mole.
The internal energy ($U$) of one mole of a monoatomic ideal gas is:
$$U = 3 \times \frac{1}{2}RT = \frac{3}{2}RT$$
The molar heat capacity at constant volume ($C_V$) is the derivative of internal energy with respect to temperature:
$$C_V = \left(\frac{\partial U}{\partial T}\right)_V = \frac{d}{dT}\left(\frac{3}{2}RT\right) = \frac{3}{2}R$$
Work done by an expanding gas is given by the integral of pressure with respect to volume: $$W = \int_{V_1}^{V_2} P dV$$
For an ideal gas, $PV = nRT$, so $P = \frac{nRT}{V}$. For an **isothermal process**, the temperature ($T$) is constant.
$$W = \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT \int_{V_1}^{V_2} \frac{1}{V} dV = nRT [\ln V]_{V_1}^{V_2}$$
$$W = nRT(\ln V_2 - \ln V_1) = nRT \ln \frac{V_2}{V_1}$$
A **free expansion** is an expansion of a gas into a vacuum. This process is highly irreversible. According to the First Law of Thermodynamics, $\Delta U = Q - W$, where $Q$ is heat added and $W$ is work done by the system.
In a free expansion, the gas does no work because it expands against a vacuum, meaning there is no external pressure ($W = 0$). Additionally, the process happens so quickly that there is no time for heat exchange with the surroundings ($Q = 0$).
Therefore, for a free expansion of an ideal gas, the change in internal energy is zero: $$\Delta U = 0$$
This implies that the temperature of the ideal gas also remains constant.
This is a direct application of the **Stefan-Boltzmann Law**. The law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's absolute temperature ($T$).
The equation is: $$\frac{P}{A} = \epsilon\sigma T^4$$
where $P$ is power, $A$ is area, $\epsilon$ is emissivity (1 for a perfect black body), and $\sigma$ is the Stefan-Boltzmann constant. Thus, the power radiated per unit area is proportional to $T^4$.
Bernoulli's equation for a streamline is typically written as:
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
The equation has three pressure-like terms:
Therefore, the term representing dynamic pressure is $\frac{1}{2}\rho v^2$.
The **magnetic moment** ($\vec{\mu}$) of a planar current loop is a vector quantity that describes the strength and orientation of the magnetic source. Its magnitude is defined as the product of the current and the area of the loop.
The magnitude is given by: $$\mu = IA$$
The question asks for a scalar value, and option B represents the magnitude of the magnetic moment. While option C is the vector form, B is the intended answer for the scalar magnitude.
Total questions: 30
Correct answers: 0
Incorrect answers: 0
Score: 0%
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