Physics Test 121

The object is always placed in front of the mirror hence the object distance is taken as negative whereas the centre of curvature and focus lie in the front of the concave mirror so that the radius of curvature and the focal length is taken as negative in the case of the concave mirror.

By using sign convention, from mirror equation, we get;

Given,

M_He= 4, mHe = 16 g

M_ox= 32, mox = 16 g

Specific heat of mixture at constant volume is given by,

Given, the radius of a non-conducting disc, R. Charge on the disc, q. Angular velocity of the disc, ω. The magnetic moment of a current loop is given by the formula,

μ=IA

where I is the current through the loop and A is the area of the loop.

Surface charge density is defined as charge per unit area. Here, area of the disc is πR² and charge is q so, surface charge density will be,

Now, we take a small cross section such that its length is dr with its inner radius is r and outer radius is r+dr and dr<<r.

Here, surface charge density for this region will be,

Now, we get the magnetic moment of the disc by integrating on both sides of the above equation from 0 to R on right side and 0 to μ on left side,

Let us consider that the work needed to take a unit mass from point P to ∞ is −V_p where V_p is the gravitational potential due to the disc. For finding V_p, we will consider a small element of thickness dr from the disc.

Now, let us take such an element at distance r from the center of the disc as shown in the figure.

Now, the mass of the small element is given by,

So, the gravitational potential due to the entire disc is given by integrating the potential due to the number of these small

elements within the limit 3R to 4R. That is,

Here, the distance of a small element from P is found using the Pythagoras theorem.

Therefore the work required to take a unit mass from point P to infinity is found to be V_p = − 2MG/7R(4√2−5).

In the given question, the following data are provided:

Initial temperature is 60∘C.

Final temperature is 40∘C.

The time required for cooling is 7 minutes.

We will apply Newton's law of cooling in this case, which is given by:

Where,

t indicates the time required for cooling.

T indicates the temperature of the body.

T_s indicates the temperature of the surrounding.

Now, we rearrange and perform integration in the equation (1) and we get:

Second case,

The initial temperature will be 40^∘C i.e., which was the final temperature of the first case.

We need to calculate the final temperature after an interval of 7 minutes.

Again, we apply Newton's law of cooling and we get from equation (2):

For a body moving with constant acceleration, the kinematics equation is s=ut+1/2at²

If the initial speed is zero, then the time taken to reach a distance s is 

i.e., t∝a^−0.5

In the case of a smooth inclined plane, a₁= g sin⁡θ. 

In the case of rough inclined plane, a₂ = g(sin⁡θ−μcos⁡θ).

Time taken to travel down the smooth inclined plane is t₁=t.

Time taken to travel down the smooth inclined plane is t₂=nt.

In the given problem,

The radius of the circular disc is 0.2 meter.

The uniform magnetic field of induction is 1/π (Wbm⁻²)

The axis of the disc is inclined to the magnetic field with an angle of 60^∘.

We need to find the magnetic flux linked with the disc.

For this, we have a formula, which gives magnetic flux linked with it:

ϕ=BAcos⁡θ……(1)

Where,

ϕ indicates magnetic flux linked with the disc.

B indicates magnetic field.

A indicates the area of the disc.

θ indicates the inclination of the axis of the disc to the magnetic field.

In the formula, we use the cosine component of the magnetic field, because we need to find the magnetic flux, i.e., it means we need to find the total number of magnetic field lines which passes through the given area. The component which lies along the axis of the disc is always the cosine component.

The radius of the circular disc is 0.2 meter.

We can calculate its area, given by the formula:

A=πr²

⇒A=π×(0.2)² m²

Now, substituting the required values in the equation (1), we get:

Spectral lines are produced by transitions of electrons within the atoms or ions. When an electron jumps from nth orbit to ground state, total  number of spectral lines are formed. Each element has its own unique line emission of the spectrum.

Since it is given in the question that hydrogen atoms absorb the light and subsequently emit radiations of ten different wavelengths.

So, number of spectral lines emitted are,

Now, using Rydberg's formula, ; where n1 and n2 are integers and n2 is always greater than n1.R is a constant called Rydberg constant.

Substituting the value of n=5 in Rydberg formula we get,

When a steady current flows in a metallic conductor of nonuniform cross-section then the drift speed is V_d = I/neA and the electric field E = I/ρA

⇒ Only current remains constant. 

In a metallic conductor of non-uniform cross-section, only the current remains constant along the entire length of the conductor.

According to the conservation of mechanical energy,

Loss in PE of the particle = Gain in elastic potential energy of the spring