A felt-covered beanbag is fired into a empty wooden crate that sits on a concrete floor and is open on one side. The beanbag is moving exactly horizontally when it strikes and sticks to the inside wall of the crate, which is lined with velcro. As a result of this impact, the crate slides backwards 0.935 meters along the concrete, which has a coefficient of kinetic friction of \mu_k = 0.480μ ​k ​​ =0.480. The beanbag has a mass of 0.354 kg and the empty crate has a mass of 3.77 kg. Calculate the speed of the beanbag vv at the moment it strikes the crate. You may neglect the effect of air resistance.

To solve this problem it is necessary to apply the expressions related to the calculation of angular acceleration in cylinders as well as the calculation of linear acceleration in these bodies.

By definition we know that the angular acceleration in a cylinder is given by

[tex]\alpha = \frac{2\mu_k g cos\theta}{r}[/tex]

Where,

[tex]\mu_k[/tex] = Coefficient of kinetic friction

g = Gravitational acceleration

r= Radius

[tex]\theta[/tex]= Angle of inclination

While the tangential or linear acceleration is given by,

[tex]a = g(sin\theta-\mu_k cos\theta)[/tex]

ANGULAR ACCELERATION, replacing the values that we have

[tex]\alpha = \frac{2\mu_k g cos\theta}{r}[/tex]

[tex]\alpha = \frac{2(0.4)(9.8) cos(30)}{10*10^{-2}}[/tex]

[tex]\alpha = 67.9rad/s[/tex]

LINEAR ACCELERATION, replacing the values that we have,

[tex]a = (9.8)(sin30-(0.4)cos(30))[/tex]

[tex]a = 1.5m/s^2[/tex]

Therefore the linear acceleration of the solid cylinder is [tex]1.5 m/s^2[/tex]

Answer:

the maximum distance of rotation can he stand without sliding is 1.77 m

Explanation:

given information:

angular velocity , ω = 1.58 rad/s

static friction, μs = 0.45

now we calculate the vertical force

N - W = 0, N is normal force and W is weight

N = W

   = m g

next, for the horizontal force we only have frictional force, thus

F(friction) = m a

μs N = m a

μs m g = m a

a = μs g,

now we have to find the acceleration which is both translation and cantripetal.

a = [tex]\sqrt{a_{t} ^{2}+a_{c} ^{2}  }[/tex]

[tex]a_{t} ^{2}[/tex] is the acceleration for translation

[tex]a_{t} ^{2}[/tex] = 0

[tex]a_{c} ^{2}[/tex] is centripetal acceleration

[tex]a_{c} ^{2}[/tex] = ω^2r

therefore,

a = [tex]\sqrt{a_{c} ^{2}  }[/tex]

  = [tex]a_{c} ^{2}[/tex]

  = ω^2r

Now, to find the radius, substitute the equation into the following formula

a = μs g

ω^2r = μs g

r = μs g / ω^2

 = (0.45 x 9.8) / (1.58)

 = 1.77 m

Answer:

Thermal energy will be equal to 784 J

Explanation:

We have given that mass of the child m = 40 kg

Height h = 2 m

Acceleration due to gravity [tex]g=9.8m/sec^2[/tex]

We have to find the thermal energy '

The thermal energy will be equal to potential energy

And we know that potential energy is given by

[tex]W=mgh=40\times 9.8\times 2=784J[/tex]

So the thermal energy will be equal to 784 J

Answer:

a) F = 2.7 10⁻¹⁴ N , b)  a = 2.97 10¹⁶ m / s²  c) θ = 14º

Explanation:

The magnetic force on the electron is given by the expression

     F = q v x B

Which can be written in the form of magnitude and the angle found by the rule of the right hand

     F = q v B sin θ

where θ is the angle between the velocity and the magnetic field

a) the maximum magnitude of the force occurs when the velocity and the field are perpendicular, therefore, without 90 = 1

     F = e v B

     F = 1.6 10⁻¹⁹ 2.40 10⁶ 7.10 10⁻²

     F = 2.73 10⁻¹⁴ N

     F = 2.7 10⁻¹⁴ N

b) Let's use Newton's second law

    F = m a

    a = F / m

    a = 2.7 10⁻¹⁴ / 9.1 10⁻³¹

    a = 2.97 10¹⁶ m / s²

The actual acceleration (a1) is a quarter of this maximum

    a1 = ¼ a

    a1 = 7.4 10¹⁵ m / s²

With this acceleration I calculate the force that is executed on the electron

     F = ma

    e v b sin θ= ma

    sin θ = ma / (e v B)

    sin θ = 9.1 10⁻³¹ 7.4 10¹⁵ / (1.6 10⁻¹⁹ 2.40 10⁶ 7.10 10⁻²)

    sin θ = 6.734 10⁻¹⁵ / 27.26 10⁻¹⁵

    sin θ = 0.2470

    θ = 14.3º

To develop this problem it is necessary to apply the concepts related to the Dopler effect.

The equation is defined by

[tex]f_i = f_0 \frac{c}{c+v}[/tex]

Where

[tex]f_h[/tex]= Approaching velocities

[tex]f_i[/tex]= Receding velocities

c = Speed of sound

v = Emitter speed

And

[tex]f_h = f_0 \frac{c}{c+v}[/tex]

Therefore using the values given we can find the velocity through,

[tex]\frac{f_h}{f_0}=\frac{c-v}{c+v}[/tex]

[tex]v = c(\frac{f_h-f_i}{f_h+f_i})[/tex]

Assuming the ratio above, we can use any f_h and f_i with the ratio 2.4 to 1

[tex]v = 353(\frac{2.4-1}{2.4+1})[/tex]

[tex]v = 145.35m/s[/tex]

Therefore the cars goes to 145.3m/s

Answer:

4341.44763 kg/m³

Explanation:

[tex]\rho'[/tex] = Actual density of cube = 1800 kg/m³

[tex]\rho[/tex] = Density change due to motion

v = Velocity of cube = 0.91c

c = Speed of light = [tex]3\times 10^8\ m/s[/tex]

Relativistic density is given by

[tex]\rho=\frac{\rho'}{\sqrt{1-\frac{v^2}{c^2}}}\\\Rightarrow \rho=\frac{1800}{\sqrt{1-\frac{0.91^2c^2}{c^2}}}\\\Rightarrow \rho=\frac{1800}{\sqrt{1-0.91^2}}\\\Rightarrow \rho=4341.44763\ kg/m^3[/tex]

The cube's density as measured by an experimenter in the laboratory is 4341.44763 kg/m³

To solve this problem, it is necessary to apply the concepts related to Newton's second Law as well as to the expression of mass as a function of Volume and Density.

From Newton's second law we know that

F= ma

Where,

m = mass

a = acceleration

At the same time we know that the density is given by,

[tex]\rho = \frac{m}{V} \rightarrow m = \rho V[/tex]

Our values are given as,

[tex]g = 9.8m/s^2[/tex]

[tex]m =0.459 kg[/tex]

D=0.242 m

Therefore the Force by Weight is

[tex]F_w = mg[/tex]

[tex]F_w = 0.459kg * 9.8m/s^2 = 4.498N[/tex]

Now the buoyant force acting on the ball is

[tex]F_B=\rho V g[/tex]

The value of the Volume of a Sphere can be calculated as,

[tex]V = \frac{4}{3} \pi r^3[/tex]

[tex]V =  \frac{4}{3} \pi (0.242/2)^3[/tex]

[tex]V = 0.007420m^3[/tex]

[tex]\rho_w = 1000kg/m^3 \rightarrow[/tex] Normal conditions

Then,

[tex]F_B=0.007420*(1000)*(9.8) = 72.716 N[/tex]

Therefore the Force net is,

[tex]F_{net} = F_B -F_w[/tex]

[tex]F_{net} = 72.716N - 4.498N =68.218 N[/tex]

Therefore the required Force is 68.218N

Answer:0

Explanation:

Given

circumference of circle is 2 m

Tension in the string [tex]T=5 N[/tex]

[tex]2\pi r=2[/tex]

[tex]r=\frac{2}{2\pi }=\frac{1}{\pi }=0.318 m[/tex]

In this case Force applied i.e. Tension is Perpendicular to the Displacement therefore angle between Tension and displacement is [tex]90^{\circ}[/tex]

[tex]W=\int\vec{F}\cdot \vec{r}[/tex]

[tex]W=\int Fdr\cos 90 [/tex]

[tex]W=0[/tex]

The work done by the ball as it travels once around the circular string is 0.

The given parameters;

The work-done by a body is the dot product of applied force and displacement.

For one complete rotation around the circumference of the circle, the displacement of the object is zero.

The work-done by the ball when its  makes a complete rotation around the circle is calculated as;

Work-done = F x r

where;

Work-done = 5 x 0 = 0

Thus, the work done by the ball as it travels once around the circular string is 0.

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Answer:

Explanation:

We shall consider direction towards left as positive Let the required velocity be v and let v makes an angle φ

Applying law of conservation of momentum along direction of original motion

m₁ v₁  - m₂ v₂ = m₂v₃ - m₁ v₄

0.132 x 1.25 - .143 x 1.14 = 1.03 cos43 x .143 - v cos θ

v cos θ = .8

Applying law of conservation of momentum along direction perpendicular to direction of original motion

1.03 sin 43 x .143 = .132 x v sinθ

v sinθ = .76

squaring and adding

v² = .76 ² + .8²

v = 1.1 m /s

Tan θ = .76 / .8

θ = 44°

Answer:

Change in momentum will be 1.8 kgm/sec

Explanation:

We have given mass of the ball m = 0.060 kg

Initial speed = 12 m /sec

And final speed = 18 m/sec

We have to find the change in momentum

Change in momentum is given by [tex]\Delta P=m(v_f-v_i)[/tex]

So [tex]\Delta P=0.060\times (18-(-12))=0.060\times 30=1.8kgm/sec[/tex] ( negative sign is due to finally opposite direction of ball )

So change in momentum will be 1.8 kgm/sec

The change of momentum of the tennis ball after being hit by the racket, reversing its direction, is calculated to be -1.8 kg*m/s.

To determine the change in momentum of a tennis ball following a bounce, one must calculate the initial and final momentum and find the difference between the two. Momentum (p) is the product of mass (m) and velocity (v), hence p = mv. The tennis ball in question has a mass of 0.06 kg. The initial momentum is 0.06 kg * 12 m/s = 0.72 kg*m/s. After it is struck by the racket, it rebounds in the opposite direction at a speed of 18 m/s giving it a momentum of 0.06 kg * -18 m/s = -1.08 kg*m/s (negative because direction changed). The change in momentum is the final momentum minus the initial momentum, hence -1.08 kg*m/s - 0.72 kg*m/s = -1.8 kg*m/s.

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Answer:

(a) [tex]\displaystyle s(t)= \frac{t^3}{3}-\frac{t^2}{2}-20t+C\ \ \ \ \forall\ \  3\leqslant t\leqslant 9[/tex]

(b) 78 m

Explanation:

Physics' cinematics as rates of change.

Velocity is defined as the rate of change of the displacement. Acceleration is the rate of change of the velocity.

[tex]\displaystyle v=\frac{ds}{dt}[/tex]

Knowing that

[tex]\displaystyle v(t)= t^2 - t - 20[/tex]

(a) To find the displacement we need to integrate the velocity

[tex]\displaystyle \frac{ds}{dt}=t^2 - t - 20[/tex]

[tex]\displaystyle ds=(t^2 - t - 20)dt[/tex]

[tex]\displaystyle s(t)= \int(t^2 - t - 20)dt=\frac{t^3}{3}-\frac{t^2}{2}-20t+C\ \ \ \ \forall \ \ \ 3\leqslant t\leqslant 9[/tex]

(b) The displacement can be found by evaluating the integral

[tex]\displaystyle d=\int_{3}^{9} (t^2 - t - 20)dt[/tex]

[tex]\displaystyle d=\left | \frac{t^3}{3}-\frac{t^2}{2}-20t \right |_3^9=\frac{45}{2}+\frac{111}{2}=78\ m[/tex]

To find displacement, we integrate the velocity function from 3 to 9. The displacement is the integral of the velocity function. For distance, we integrate the absolute value of the velocity function from 3 to 9 because distance is a scalar quantity and includes total path length.

To solve the problem, we need to find the displacement and distance traveled by the particle. For part (a), the displacement for the given time period is obtained by integrating the velocity function from 3 to 9. The displacement is the integral of the velocity function v(t) = t2 - t - 20 from 3 to 9, which gives us the value s(9) - s(3).

For part (b), to calculate the distance traveled, we need to integrate the absolute value of the velocity function because the distance is always positive. Negative values would represent backward motion, but the distance traveled includes total path length and does not care about direction.

So, the distance traveled from time 3 to 9 would be ∫ |t2 - t - 20| dt from 3 to 9. The calculation of this integral will give the distance traveled.

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To solve this problem it is necessary to use three key concepts. The first of these is the Centripetal Force which is in the upward direction and would be described as

[tex]F_c = \frac{mv^2}{r}[/tex]

Where,

m= mass

v= Velocity

r = Radius

The second is the voltage generated which is given by 1400N. Finally the third is the force generated by the weight and that would be described by Newton's second law as

F =mg

Where,

m = Mass

g = Acceleration gravity

F = 80*9.8

F = 784N

For balance to exist, the sum of Force must be equal to the Centripetal Force, therefore

[tex]\sum F = F_c[/tex]

[tex]T - mg = \frac{mv^2}{r}[/tex]

Replacing we have

[tex]1400 - 784 = \frac{(80kg)v^2}{5,5}[/tex]

[tex]v^2 = \frac{ 616*5.5}{80}[/tex]

[tex]v = \sqrt{42.35}[/tex]

[tex]v=6.5m/s[/tex]

Therefore the maximum speed he can tolerate at the lowest point of his swing is 6.5m/s

Answer:

The number of capillaries is

[tex]N=2.625x10^9[/tex] Capillaries

Explanation:

[tex]v_{aorta}=42cm/s[/tex], [tex]r_{aorta}=1.1 cm[/tex], [tex]v_{cap}=0.064cm/s[/tex], [tex]r_{cap}=5.5x10^{4}cm[/tex],

To find the number of capillaries in the human body use the equation:

[tex]N_{cap}=\frac{v_{aorta}*\pi*r_{aorta}^2}{v_{cap}*\pi*r_{cap}^2}[/tex]

So replacing numeric

[tex]N_{cap}=\frac{42cm/s*\pi*(1.1cm)^2}{0.064cm/s*\pi*(5.5x10^{-4}cm)^2}[/tex]

Now we can find the number of capillaries

[tex]N=26250000000[/tex]

[tex]N=2.625x10^9[/tex] Capillaries

Answer:

 i = 0.3326 L

Explanation:

A fixed string at both ends presents a phenomenon of standing waves, two waves with the same frequency that are added together. The expression to describe these waves is

    2 L = n λ           n = 1, 2, 3…

The first harmonic or leather for n = 1

Wave speed is related to wavelength and frequency

     v = λ f

     λ = v / f

Let's replace in the first equation

    2 L = 1 (v / f₁)

For the shortest length L = L-l

   2 (L- l) = 1 (v / f₂)

These two equations form our equation system, let's eliminate v

    v = 2L f₁

    v = 2 (L-l) f₂

    2L f₁ = 2 (L-l) f₂

    L- l = L f₁ / f₂

    l = L - L f₁ / f₂

    l = L (1- f₁ / f₂)

.

Let's calculate

    l / L = (1- 309/463)

    i / L = 0.3326

Answer: 75ºC

Explanation:

Assuming that the Styrofoam is perfectly adiabatic, and neglecting the effect of the sugar on the system, the heat lost by the tea, can only be transferred to the spoon, reaching all the system to a final equilibrium temperature.

If the heat transfer process is due only to conduction, we can use this empirical relationship for both objects:

Qt = ct . mt . (tfn – ti)

Qs = cs . ms . (ti – tfn)

If the cup is perfectly adiabatic, it must be Qt = Qs

Using the information provided, and solving for tfinal, we get:

tfinal = (83,600 + 1,800) / (90 + 1045) ºC  

tfinal = 75º C

Answer:

[tex]I_{edge} = 6.67 kg.m^2[/tex]

Explanation:

given,

mass of rod = 5 Kg

Length of rod = 2 m

R = 1 m

moment of inertial from one edge of the rod = ?

moment of inertia of rod through center of mass

[tex]I_{CM}= \dfrac{1}{12}M(L)^2[/tex]

using parallel axis theorem

[tex]I_{edge} = I_{CM} + MR^2[/tex]

[tex]I_{edge} =\dfrac{1}{12}ML^2+ M(\dfrac{L}{2})^2[/tex]

[tex]I_{edge} =\dfrac{1}{12}ML^2+ \dfrac{ML^2}{4}[/tex]

[tex]I_{edge} =\dfrac{1}{3}ML^2[/tex]

now, inserting all the given values

[tex]I_{edge} =\dfrac{1}{3}\times 5 \times 2^2[/tex]

[tex]I_{edge} = 6.67 kg.m^2[/tex]

The x-coordinate of the center of mass of the lidless shoebox's base, which is a rectangle, will be at half its length. Therefore, the x-coordinate is 24 cm.

The center of mass is the coordinate that averages the distribution of mass in a physical object. To calculate the center of mass, one must consider the dimensions of the object. For a rectangular object like the shoebox's base with length, L = 48 cm, and width, W = 25 cm, the x-coordinate of the center of mass will be half the length. Therefore, the x-coordinate of the center of mass of the shoebox's base is L/2 = 48 cm / 2 = 24 cm. This is because in this rectangular coordinate system, the center of mass is located at the middle along each dimension (L, W).

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The x-coordinate of the center of mass of the shoebox's base is half its length which is 24 cm.

The center of mass of an object is basically the average position of all its parts. Given that the shoebox's base lies on the xy-plane, the x-coordinate for its center of mass will simply be half the length of the base. Hence, using the length L = 48 cm, the x-coordinate for the center of mass is: L/2 = 48/2 = 24 cm. Therefore, the center of mass of the shoebox's base along the x-axis lies at 24 cm.

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Answer:

T_t = Ts  (1-A[tex])^{1/4}[/tex] √ (Rs/D)

Explanation:

The black body radiation power is given by Stefan's law

       P = σ A e T⁴

This power is distributed over a spherical surface, so the intensity of the radiation is

     I = P / A

Let's apply these formulas to our case. Let's start by calculating the power emitted by the Sun, which has an emissivity of one (e = 1) black body

    P_s = σ A_s 1 T_s⁴

This power is distributed in a given area, the intensity that reaches the earth is

     I = P_s / A

    A = 4π R²

The distance from the Sun Earth is R = D

     I₁ = Ps / 4π D²

     I₁ = σ (π R_s²) T_s⁴ / 4π D²

     I₁ = σ T_s⁴ R_s² / 4D²

Now let's calculate the power emitted by the earth

     P_t = σ A_t (e) T_t⁴

     I₂ = P_t / A_t

     I₂ = P_t / 4π R_t²2

     I₂ = σ (π R_t²) T_t⁴ / 4π R_t²2

     I₂ = σ T_t⁴ / 4

The thermal equilibrium occurs when the emission of the earth is equal to the absorbed energy, the radiation affects less the reflected one is equal to the emitted radiation

     I₁ - A I₁ = I₂

     I₁ (1 - A) = I₂

Let's replace

    σ T_s⁴  R_s²/4D²   (1-A)  = σ T_t⁴ / 4

    T_s⁴ R_s² /D²   (1-A) = T_t⁴

    T_t⁴ = T_s⁴  (1-A)  (Rs / D) 2

    T_t = Ts  (1-A[tex])^{1/4}[/tex] √ (Rs/D)

Answer:

[tex]\dfrac{F}{l}=2\times 10^{-7}\ N/m[/tex]

Explanation:

It is given that,

Distance between two infinitely long wires, d = 1 m

Current flowing in both of the wires, I = 1 A

The magnetic field in a wire is given by :

[tex]B=\dfrac{\mu_oI}{2\pi d}[/tex]

The force per unit length acting on the two infinitely long wires is given by :

[tex]\dfrac{F}{l}=\dfrac{\mu_o I_1I_2}{2\pi d}[/tex]

[tex]\dfrac{F}{l}=\dfrac{4\pi \times 10^{-7}\times 1^2}{2\pi \times 1}[/tex]

[tex]\dfrac{F}{l}=2\times 10^{-7}\ N/m[/tex]

So, the force acting on the parallel wires is [tex]2\times 10^{-7}\ N/m[/tex]. Hence, this is the required solution.

The force per unit length between the two wires is [tex]\( 2 \times 10^{-7} \)[/tex] newtons per meter.

The force per unit length (F/L) between two infinitely long wires carrying current can be calculated using Ampere force law, which is given by:

[tex]\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi r} \][/tex]

 Plugging in the values, we get:

[tex]\[ \frac{F}{L} = \frac{(4\pi \times 10^{-7} \text{ N/A}^2) \times (1 \text{ A}) \times (1 \text{ A})}{2 \pi \times (1 \text{ m})} \][/tex]

Simplifying the expression by canceling out [tex]\( \pi \)[/tex] and multiplying the numerical values, we have:

[tex]\[ \frac{F}{L} = \frac{(4 \times 10^{-7} \text{ N/A}^2) \times (1 \text{ A})^2}{2 \times (1 \text{ m})} \][/tex]

[tex]\[ \frac{F}{L} = \frac{4 \times 10^{-7} \text{ N}}{2 \text{ m}} \][/tex]

[tex]\[ \frac{F}{L} = 2 \times 10^{-7} \text{ N/m} \][/tex]

 Therefore, the force per unit length between the two wires is [tex]\( 2 \times 10^{-7} \)[/tex] newtons per meter.

The answer is: [tex]2 \times 10^{-7} \text{ N/m}[/tex]

Answer:

Total internal reflection is going on. The refractive index of water is about 1.3, so sin 90/sin r=1/sinr=1.3. So the fish can only see objects outside the water within about 50 degrees of the vertical

Answer:

the heat released by the magnesium is 72 kJ

Explanation:

the heat exchanged will be

Q = m*c*(T final - T initial)  

where Q= heat released, c= specific heat capacity, T initial= initial temperature of water, T final = final temperature of water

Assuming the specific heat capacity of water as c= 1 cal/g°C=4.186 J/g°C

replacing values

Q = m * c* (T final - T initial) = 1910 g * 4.186 J/g°C*(33.10 °C - 24°C) =  72 kJ

I examine the same second hand on the clock. Again, there are two points called A and B on the clock, with A farther from the center than B. "The angular acceleration for both points is 0" is true.

Answer: Option C

Explanation:

The clock is in rotator motion. All the three hands of clock, move in same direction, but different speeds. And hence, we count hours, minutes and seconds.  And, when we take each hand, they move related to the centre of the clock, where all the three are attached.

So, there is a centripetal acceleration which depends upon the velocity.  But, the motion is uniform everywhere in the circle. The hands have no tangential acceleration. And hence, there is no angular acceleration, which is derived from tangential one. So, at any point, the angular acceleration is zero.

Answer:

a) Fh = 25.23 N

b) The direction of the force exerted by the hand is pointed to the downward (negative) direction.

c) Fs = 37.97 N

d) The direction of the force exerted by the shoulder is pointed to the upward (positive) direction.

Explanation:

Given data

Distance from the small end of the bat to the shoulder: d = 23.5 cm

Distance from the small end of the bat to the center of mass: r = 70.0 cm

Mass of the bat: m = 1.30 Kg

This situation can be seen in the pic.

a) We can apply

∑τA = 0   ⇒  + (23.5cm)*Fh - (70 - 23.5)cm*m*g = 0

⇒   + (23.5cm)*Fh - (70 - 23.5)cm*(1.30 Kg)*(9.8 m/s²) = 0

⇒   Fh = 25.23 N (↓)

b) The direction of the force exerted by the hand is pointed to the downward (negative) direction.

c) We apply

∑Fy = 0  (↑)

⇒      - Fh + Fs - m*g = 0

⇒     Fs = Fh + m*g

⇒     Fs = 25.23 N + (1.30 Kg)*(9.8 m/s²)

⇒     Fs = 37.97 N (↑)

d) The direction of the force exerted by the shoulder is pointed to the upward (positive) direction.

Answer: I = k^2m.. equa1

I = moment of inertia

M = mass of skater

K = radius of gyration.

When her angular momentum is conserved we have

Iw = I1W1... equ 2

Where I = with extended arm, w = angular momentum =40rads/s, I1 = inertia when hands her tucked in, w1 = angular momentum when hands are tucked in.

Substituting equation 1 into equ2 and simplifying to give

W = (k/k1)^2W..equation 3

Where'd k= 64cm, k1 = 32cm, w = angular momentum when hands is tucked in= 40rad/s

Substituting figures into equation 3

W1 = 10rad/s

Explanation:

Assuming a centroidal axis of the skater gives equation 1

Final answer:

When the moment of inertia is doubled, the angular velocity will decrease by half due to the conservation of angular momentum.

Explanation:

When the moment of inertia is doubled, the angular velocity will decrease by half due to the conservation of angular momentum. In this case, Kristen's initial angular velocity is 40 rad/s, and her initial moment of inertia is 32 cm. After doubling her radius of gyration to 64 cm, her final moment of inertia is 128 cm. Using the conservation of angular momentum equation, we can calculate her final angular velocity:

Initial Angular Momentum = Final Angular Momentum

Initial Angular Velocity * Initial Moment of Inertia = Final Angular Velocity * Final Moment of Inertia

Substituting the values: 40 rad/s * 32 cm = Final Angular Velocity * 128 cm

Simplifying the equation: Final Angular Velocity = 10 rad/s

Therefore, Kristen's angular velocity about her longitudinal axis after increasing her radius of gyration is 10 rad/s.

Answer:C

Explanation:

Given

Four masses are attached to the wire such that distance between two mass is L

therefore the Length of wire is 4 L

and the center of mass is at 2L

moment of inertia is distribution of mass from its rotational axis

thus moment of Inertia I is given by

[tex]I=m\times (\frac{3L}{2})^2+m\times (\frac{L}{2})^2+m\times (\frac{3L}{2})^2+m\times (\frac{L}{2})^2[/tex]

[tex]I=2\times m\times (\frac{L}{2})^2+2\times m\times (\frac{3L}{2})^2[/tex]

[tex]I=\frac{2mL^2}{4}+\frac{18mL^2}{4}[/tex]

[tex]I=5mL^2[/tex]

The rotational inertia of the four-point mass system about its center of mass, with equal spacing between adjacent masses, would be 2mℓ^2.

The rotational inertia of a system around any particular axis is given by sum of the product of the mass of each particle and square of perpendicluar distance from the axis of rotation. In the case of our four-point mass system, the masses are arranged such that they are all equidistant from the center. Therefore, the total inertia, Icm, is the sum of the inertias of each individual mass.

Assuming the center of rotation is halfway between the second and third mass, there will be two masses at distance ℓ/2 and two at 3ℓ/2. Therefore, Icm= 2* m*((ℓ/2)^2) + 2* m*((3ℓ/2)^2) = 2mℓ^2. Hence, the correct answer is E.) 2mℓ^2.

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Answer:

C. At the bottom of the circle.

Explanation:

Lets take

Radius of the circle = r

Mass = m

Tension = T

Angular speed = ω

The radial acceleration towards = a

a= ω² r

Weight due to gravity = mg

T - m g = m a

T =  m ω² r  + m g

T + m g = m a

T=  m ω² r -m g

From above equation we can say that tension is grater when ball at bottom of the vertical circle.

Therefore the answer is C.

C. At the bottom of the circle.

Final answer:

The tension in the cord connected to a ball whirled in a vertical circle is greatest at the bottom of the circle because the tension must overcome gravity and provide the centripetal force to keep the ball in motion.

Explanation:

The question asks, "A child whirls a ball in a vertical circle. Assuming the speed of the ball is constant (an approximation), when would the tension in the cord connected to the ball be greatest?" The correct answer is c. At the bottom of the circle.

When the ball is at the bottom of the circle, the tension in the string is highest because it must counteract both the gravitational force pulling downwards and provide enough centripetal force to keep the ball moving in a circular path. At this point, the sum of the gravitational force and the centripetal force dictates the necessary tension, making it greater than at any other point in the ball's circular trajectory. This is in contrast to the top of the circle, where the tension is lowest since gravity assists in providing the centripetal force needed, sometimes reducing the tension in the string to nearly zero if the ball moves at the minimum speed required to continue in circular motion.

Answer with Explanation:

Mass of block=1.1 kg

Th force applied on block is given by

F(x)=[tex](2.4-x^2)\hat{i}N[/tex]

Initial position of the block=x=0

Initial velocity of block=[tex]v_i=0[/tex]

a.We have to find the kinetic energy of the block when it passes through x=2.0 m.

Initial kinetic energy=[tex]K_i=\frac{1}{2}mv^2_i=\frac{1}{2}(1.1)(0)=0[/tex]

Work energy theorem:

[tex]K_f-K_i=W[/tex]

Where [tex]K_f=[/tex]Final kinetic energy

[tex]K_i[/tex]=Initial kinetic energy

[tex]W=Total work done[/tex]

Substitute the values then we get

[tex]K_f-0=\int_{0}^{2}F(x)dx[/tex]

Because work done=[tex]Force\times displacement[/tex]

[tex]K_f=\int_{0}^{2}(2.4-x^2)dx[/tex]

[tex]K_f=[2.4x-\frac{x^3}{3}]^{2}_{0}[/tex]

[tex]K_f=2.4(2)-\frac{8}{3}=2.13 J[/tex]

Hence, the kinetic energy of the block as it passes thorough x=2 m=2.13 J

b.Kinetic energy =[tex]K=2.4x-\frac{x^3}{3}[/tex]

When the kinetic energy is maximum then [tex]\frac{dK}{dx}=0[/tex]

[tex]\frac{d(2.4x-\frac{x^3}{3})}{dx}=0[/tex]

[tex]2.4-x^2=0[/tex]

[tex]x^2=2.4[/tex]

[tex]x=\pm\sqrt{2.4}[/tex]

[tex]\frac{d^2K}{dx^2}=-2x[/tex]

Substitute x=[tex]\sqrt{2.4}[/tex]

[tex]\frac{d^2K}{dx^2}=-2\sqrt{2.4}<0[/tex]

Substitute x=[tex]-\sqrt{2.4}[/tex]

[tex]\frac{d^2K}{dx^2}=2\sqrt{2.4}>0[/tex]

Hence, the kinetic energy is maximum at x=[tex]\sqrt{2.4}[/tex]

Again by work energy theorem , the  maximum kinetic energy of the block between x=0 and x=2.0 m is given by

[tex]K_f-0=\int_{0}^{\sqrt{2.4}}(2.4-x^2)dx[/tex]

[tex]k_f=[2.4x-\frac{x^3}{3}]^{\sqrt{2.4}}_{0}[/tex]

[tex]K_f=2.4(\sqrt{2.4})-\frac{(\sqrt{2.4})^3}{3}=2.48 J[/tex]

Hence, the maximum energy of the block between x=0 and x=2 m=2.48 J

Answer:

(a) 498.4 Hz

(b) 442 Hz

Solution:

As per the question:

Length of the wire, L = 1.80 m

Weight of the bar, W = 531 N

The position of the copper wire from the left to the right hand end, x = 0.40 m

Length of each wire, l = 0.600 m

Radius of the circular cross-section, R = 0.250 mm = [tex].250\times 10^{- 3}\ m[/tex]

Now,

Applying the equilibrium condition at the left end for torque:

[tex]T_{Al}.0 + T_{C}(L - x) = W\frac{L}{2}[/tex]

[tex]T_{C}(1.80 - 0.40) = 531\times \frac{1.80}{2}[/tex]

[tex]T_{C} = 341.357\ Nm[/tex]

The weight of the wire balances the tension in both the wires collectively:

[tex]W = T_{Al} + T_{C}[/tex]

[tex]531 = T_{Al} + 341.357[/tex]

[tex]T_{Al} = 189.643\ Nm[/tex]

Now,

The fundamental frequency is given by:

[tex]f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex]

where

[tex]\mu = A\rho = \pi R^{2}\rho[/tex]

(a) For the fundamental frequency of Aluminium:

[tex]f = \frac{1}{2L}\sqrt{\frac{T_{Al}}{\mu}}[/tex]

[tex]f = \frac{1}{2L}\sqrt{\frac{T_{Al}}{\pi R^{2}\rho_{Al}}}[/tex]

where

[tex]\rho_{l} = 2.70\times 10^{3}\ kg/m^{3}[/tex]

[tex]f = \frac{1}{2\times 0.600}\sqrt{\frac{189.643}{\pi 0.250\times 10^{- 3}^{2}\times 2.70\times 10^{3}}} = 498.4\ Hz[/tex]

(b)  For the fundamental frequency of Copper:

[tex]f = \frac{1}{2L}\sqrt{\frac{T_{C}}{\mu}}[/tex]

[tex]f = \frac{1}{2L}\sqrt{\frac{T_{C}}{\pi R^{2}\rho_{C}}}[/tex]

where

[tex]\rho_{C} = 8.90\times 10^{3}\ kg/m^{3}[/tex]

[tex]f = \frac{1}{2\times 0.600}\sqrt{\frac{341.357}{\pi 0.250\times 10^{- 3}^{2}\times 2.70\times 10^{3}}} = 442\ Hz[/tex]

Answer:

2.2

Explanation:

[tex]Q_u[/tex] = Heat rejection in the condenser = 3300 kJ/h

[tex]Q_L[/tex] = Heat gain to the food department = 4800 kJ/h

Power output is given by

[tex]W=Q_u-Q_L\\\Rightarrow W=4800-3300\\\Rightarrow W=1500\ kJ/h[/tex]

COP of a refrigerator is given by

[tex]COP=\frac{Desired\ effect}{Work}\\\Rightarrow COP=\frac{Q_L}{W}\\\Rightarrow COP=\frac{3300}{1500}\\\Rightarrow COP=2.2[/tex]

The COP of the refrigerator is 2.2

Answer:

0.639 V

Explanation:

The volatge of the cell containing both Ag/AgCl reference electrode and

[tex]Fe^{2+}/Fe^{3+}[/tex] electrode = 0.693 V

Thus,

[tex]E_{cathod}-E_{anode} =0.693 V[/tex]

E_{anode}=0.197 V

Note: potential of the silver-silver chloride reference electrode (0.197 V)

⇒E_{Cthode}= 0.693+0.197 = 0.890V

To calculate the voltage of the cell containing both the calomel reference

electrode and [tex]Fe^{2+}/Fe^{3+}[/tex] electrode as follows

Voltage of the cell = [tex]E_{cathod}-E_{anode}[/tex]

E_{anode}= calomel electrode= 0.241 V

Voltage of the cell = 0.890-0.241 = 0.639 V

Therefore, the new volatge is = 0.639 V