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1. The formula for power: Power (P) = Force (F) × Velocity (v)

2. Reduced by half:

1. New Force (F′) = $\frac{F}{2}$
2. New Velocity (v′) = v/2 $\frac{v}{2}$

3. Calculate the new power (P′):

1. P′=F′×v′
2. P′=$\frac{f}{2}$ × $\frac{v}{2}$
3. P′=$\frac{1}{4}*FV$

4. Compare to original power: Since P=Fv, then P′= $\frac{1}{4}*P$ . The power is reduced to a quarter.

A bimetallic strip is made of two metals with different coefficients of linear expansion. When heated, the metal with the higher linear expansivity expands more than the other. This unequal expansion causes the strip to bend, with the more expansive metal on the outside of the curve. This principle is used in thermostats and thermometers

This is a calorimetry problem based on the principle of heat exchange: Heat lost by hot object = Heat gained by cold objects (water and calorimeter).

1. Given values:

1. Mass of metal (mm​) = 0.5 kg
2. Initial temp of metal (Tm,i​) = 100°C
3. Final mixture temp (Tf​) = 25°C
4. Initial temp of water/calorimeter (Tw,i​,Tcal,i​) = 15°C
5. Heat capacity of water (Cw​) = $420 J K^{-1}$ (This is the total heat capacity, not specific)
6. Heat capacity of calorimeter (Ccal​) = $80 J K^{-1}$ (Total heat capacity)

2. Calculate temperature changes:

1. Temperature change of metal (ΔTm​) = Tm,i​−Tf​=100∘C−25∘C=75∘C=75 K
2. Temperature change of water (ΔTw​) = Tf​−Tw,i​=25∘C−15∘C=10∘C=10 K
3. Temperature change of calorimeter (ΔTcal​) = Tf​−Tcal,i​=25∘C−15∘C=10∘C=10 K

3. Apply heat exchange equation:

1. Heat Lost by Metal = (Heat Gained by Water) + (Heat Gained by Calorimeter)
2. mm​cm​ΔTm​=Cw​ΔTw​+Ccal​ΔTcal​

4. Substitute values and solve for cm​ (specific heat capacity of the metal):

1. 0.5×cm​×75=(420×10)+(80×10)
2. 37.5cm​=4200+800
3. 37.5cm​=5000
4. cm​=$\frac{5000}{37.5}≈133.33 J kg^{-1} K^{-1}$

The principle of conservation of linear momentum states that in the absence of external forces, the total momentum of a system remains constant. A rocket uses this principle by expelling hot gases at high velocity in one direction. To conserve the total momentum of the rocket-gas system, the rocket itself gains momentum in the opposite direction, propelling it forward.

Isotopes of an element are atoms that have the same number of protons (atomic number) but a different number of neutrons. Because they have the same number of protons, they belong to the same chemical element. The difference in the number of neutrons results in different mass numbers. For example, Carbon-12 and Carbon-14 are isotopes of carbon, both having 6 protons, but Carbon-12 has 6 neutrons while Carbon-14 has 8 neutrons.

The half-life is the time it takes for half of the radioactive atoms in a sample to decay. In 30 years, there will have been 3 half-lives (30 years / 10 years/half-life = 3 half-lives). After the first half-life, 1/2 of the sample remains. After the second, 1/2 of that remains, which is 1/4 of the original. After the third half-life, 1/2 of the 1/4 remains, which is 1/8 of the original sample

When the elevator is stationary, there is no acceleration. According to Newton's first law, the net force on the person is zero. The forces acting on the person are their weight (W) acting downwards and the normal force (N) from the scale acting upwards. Therefore, $N - W = 0$ , which means $N = W$ . The reading on the scale, which is the normal force, is equal to the person's normal weight

In an enclosed space, such as a large hall or a stadium, sound waves can reflect off the walls, ceiling, and other surfaces. When these reflections arrive after the direct sound, they are called echoes. When the reflections are very rapid and merge with the original sound, it is called reverberation. Both of these phenomena interfere with the clarity of speech and music, making it difficult to understand what is being said.

1. General wave equation: For a wave moving in the positive x-direction is y=Asin(kx−ωt), where A is amplitude, k is wave number, and ω is angular frequency.

2. Extract given values:

1. Amplitude (A) = 0.15 m
2. Frequency (f) = 550 Hz
3. Wavelength (λ) = 0.01 m

1. Calculate wave speed (v):
2. v=fλ=550 Hz×0.01 m=5.5 m/s

3. Calculate angular frequency (ω):

1. ω=2πf=2π×550=1100π rad/s

4. Calculate wave number (k):

1. k=$\frac{2π}{λ}=\frac{2π}{0.01}=200π rad/m$

5. Substitute values into the general equation:

1. y=0.15sin(200πx−1100πt)

6. Factor out 200π to match option A's format (y=Asin[k(x−vt)], where k=200π and v=5.5):

1. y=0.15sin[200π(x−(1100π/200π)t)]
2. y=0.15sin[200π(x−5.5t)]

1. Identify given values:

1. Mass (m) = 500g = 0.5 kg
2. Extension (x) = 20cm = 0.2 m
3. Final velocity (v) = 40 m/s

2. Apply Work-Energy Theorem: The work done by the catapult equals the change in kinetic energy of the stone.

1. Work Done (W) = Force (F) × Distance (x)
2. Kinetic Energy (KE) = $\frac{1}{2}mv^2$
3. Therefore, $F*X=\frac{1}{2}mv^2$

3. Substitute values and solve for F:

1. $F*0.2=\frac{1}{2}*0.5*40^2$
2. F×0.2=0.25×1600
3. F×0.2=400
4. $F=\frac{400}{0.2}=2000 N$
5. $\therefore$ F=2.0×10³ N

A vector quantity is a physical quantity that has both magnitude and direction. Momentum is a vector quantity, as it is the product of mass (a scalar) and velocity (a vector), and it has a direction. Speed, mass, and work are all scalar quantities, having only magnitude.

Getting the value

1. Velocity Ratio (VR) = 5
2. Efficiency $n = 75\% = \frac{75}{100} = 0.75$
3. Load = 150 N

Find Mechanical Advantage (MA):

1. $η=\frac{MA}{VR}​⇒MA=η⋅VR=0.75 ⋅ 5=3.75$

Find Effort:

1. $Effort=\frac{Load}{MA}​=\frac{150​}{3.75}=40N$

The Joule (J) is the SI unit of energy and work. The Watt (W) is the unit of power (energy per unit time). The Newton (N) is the unit of force, and the Pascal (Pa) is the unit of pressure.

This is the definition of inertia, which is the property of a body to resist changes in its state of motion. It is the basis for Newton's first law of motion. A more massive body has more inertia

An optical fibre is a device that transmits light over long distances by making use of the principle of total internal reflection. The fibre consists of a core with a high refractive index surrounded by a cladding with a lower refractive index. When light enters the core at a shallow angle, it is reflected repeatedly off the boundary between the core and cladding, staying within the fibre as it travels.

The energy stored in a capacitor (E) is given by the formula $E = \frac{1}{2}CV^2$ , where $ is the capacitance and $V$ is the potential difference. The capacitance of a parallel plate capacitor is given by $C = \frac{\epsilon A}{d}$ , which is constant if the area (A) and separation ($d$) are constant. If the potential difference (V) is doubled, the new potential difference is 2V. The new energy stored will be $E_{new} = \frac{1}{2}C(2V)^2 = \frac{1}{2}C(4V^2) = 4(\frac{1}{2}CV^2) = 4E$ . Therefore, the energy stored is quadrupled.

The horizontal and vertical motions of a projectile are independent of each other. The time it takes for the body to hit the ground is determined by the vertical motion. The equation for vertical motion is $h = ut + \frac{1}{2}gt^2$ . Since the body is projected horizontally, the initial vertical velocity (u) is 0 m/s. So, the equation simplifies to $h = \frac{1}{2}gt^2$ . Rearranging to solve for $t$ , we get $t^2 = 2h/g$ . Substituting the values, $t^2 = (2 \times 80) / 10 = 160 / 10 = 16$ . Therefore, $t = \sqrt{16} = 4$ s.

Total Internal Reflection (TIR) is a phenomenon where a light ray is completely reflected back into the first medium. This occurs only under two specific conditions:

1. The light must travel from an optically denser medium (higher refractive index) to an optically rarer medium (lower refractive index).
2. The angle of incidence must be greater than the critical angle for that specific interface.

Radiation is the transfer of heat through electromagnetic waves, and it does not require a material medium. The space between the Sun and the Earth is a vacuum, so heat can only be transferred by radiation. Conduction and convection require a medium for transfer

Most liquids expand when heated and contract when cooled. However, water behaves anomalously between 0°C and 4°C. When heated from 0°C to 4°C, water contracts and its density increases, reaching its maximum density at 4°C. When heated further above 4°C, it expands like a normal liquid