Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.
x = 6 cos t, y = 6 sin t; 0 ≤ t ≤ 2π

A. x^2 - y^2 = 6; -6 ≤ x ≤ 6
B. x^2 - y^2 = 36; -6 ≤ x ≤ 6
C. x^2 + y^2 = 36; -6 ≤ x ≤ 6

Pressure is the force per unit area. It is calculated by dividing the force applied on an object by the area over which the force is distributed.

Pressure is defined as force per unit area.

Pressure is calculated by dividing the force applied on an object by the area over which the force is distributed.

For example, if you push down on a table with a force of 50 Newtons over an area of 2 square meters, the pressure on the table is 25 Newtons per square meter.

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Answer:

[tex]W = 365.7 J[/tex]

Explanation:

As we know that work done is defined as the product of force and its displacement in the direction of the force

So here we know that

applied force by Julie is same as that of weight of her suitcase

So we will have

[tex]F = mg[/tex]

[tex]F = (8\times 9.81)[/tex]

[tex]F = 78.48 N[/tex]

now we know that the total height to which the suitcase is raised is given as

[tex]H = L sin\theta[/tex]

[tex]H = 18 sin15[/tex]

[tex]H = 4.66 m[/tex]

now the work done is given as

[tex]W = F.d[/tex]

[tex]W = 78.48(4.66)[/tex]

[tex]W = 365.7 J[/tex]

Explanation of how a lever works and how to calculate its mechanical advantage when lifting a load with a given input force.

A lever is a simple machine that consists of a rigid bar pivoted at a fixed point called the fulcrum. It helps in exerting a smaller force over a longer distance to lift a heavier load over a shorter distance.

The mechanical advantage of a lever is calculated by dividing the length of the effort arm by the length of the load arm (MA = Le/Lr). It tells how many times a lever multiplies the input force to lift a load.

In the described scenario, with an input force of 500 N and a load of 2000 N, the mechanical advantage of the lever can be calculated using the formula MA = Load / Effort, which will be 4.

Answer:

C. Mercury → Mars → Neptune → Uranus      

Explanation:

There are 8 planets in the solar system. The smallest planet is mercury and the largest planet is Jupiter. There are 4 rocky planets and 4 gaseous planets.

The arrangement of planets in increasing of their diameter is:

Mercury

Mars

Venus

Earth

Neptune

Uranus

Saturn

Jupiter

Thus, from the given options, the correct sequence is:

C. Mercury → Mars → Neptune → Uranus  

By definition, the law of conservation of energy states that:

Ei = Ef

Where,

Ei: initial energy

Ef: final energy

Therefore, no matter the type of energy, always the final energy is equal to the final energy.

Energy can be transformed into another type of energy. For example, the potential energy can be transformed into kinetic energy.

Also, energy is not created, nor destroyed.

Answer:

The following is not true about the Law of Conservation of Energy:

A. It states that the total energy in the universe keeps increasing.

Casey can organize physics formulas by categorizing them based on topics, arranging them sequentially, using color-coded tabs, prioritizing key formulas, and creating a concise index for quick reference during the open book test.

Casey can adopt a systematic approach to organize his physics formulas for the upcoming open book test, promoting efficiency and accessibility during the examination. Firstly, he can categorize the formulas based on the topics or chapters they belong to. This helps create a structured framework that aligns with the course syllabus, making it easier for Casey to locate specific formulas relevant to different sections of the test.

Within each category, Casey may further arrange the formulas in a logical sequence, following the order of topics covered in the class. This sequential arrangement aids in maintaining a smooth flow while navigating through the formulas during the test, reducing the time spent searching for the right information. Additionally, he could use color-coded tabs or highlighters to distinguish between different formula categories, providing a visual aid for quick identification.

Considering the importance of some formulas over others, Casey might prioritize key formulas or those frequently used in class. By placing these prominently at the beginning of each category, he ensures immediate access to crucial information. Moreover, creating a concise index or summary at the front of his formula compilation can serve as a quick reference guide, aiding Casey in quickly identifying the page numbers corresponding to specific formulas.

By implementing these strategies, Casey can enhance his organization of physics formulas, facilitating a more efficient and stress-free experience during the open book test.

The question probable maybe: What organizational method could Casey employ to effectively categorize and arrange the various physics formulas for his open book test, ensuring easy access and streamlined use during the examination?

Final answer:

Snell's Law is the physical principle that defines the relationship between the angles of incidence and refraction when light transitions between two different media, mathematically expressed as n1 sin θ1 = n2 sin θ2.

Explanation:

Snell's Law, also known as the law of refraction, describes the relationship between the angles of incidence and refraction when a light ray passes through the interface between two different media. This fundamental physical principle is expressed mathematically as n1 sin θ1 = n2 sin θ2, where 'n1' and 'n2' are the indices of refraction of the first and second media, respectively, and 'θ1' and 'θ2' are the angles of incidence and refraction. Snell's Law is crucial for understanding phenomena such as the bending of light, and it has applications in fields like optics and engineering.

Answer:

a. 2.545 *10^8 J

b. 2.545 *10^8 J

c. 2.545 *10^8 J

d. 141.688 m/s

Explanation:

Data:

mass, m =  2.5*10^4 kg

Force, F = 5.00*10^5 N

distance, d = 509 m

a. Work, W = F*d = 5.00*10^5 N * 509 m = 2.545 *10^8 J

b. Change in kinetic energy, ΔKE = W = 2.545 *10^8 J

c. Change in kinetic energy = final kinetic energy - initial kinetic energy

If the train started from rest, initial kinetic energy = 0

Then, final kinetic energy, KEf = ΔKE = 2.545 *10^8 J

d. From the definition of kinetic energy we can get the final speed (vf) as follows:

KEf = (1/2)*m*vf^2

vf = √(KEf*2/m)

vf = √(2.545 *10^8*2/2.5*10^4 )

vf = 141.688 m/s

Explanation:

Given that,

Force of gravity acting on the person, F = mg = 850 N

The roof is on the 20 degrees slope with the surface. We need to find the magnitude of normal force of the roof on the worker. It can be calculated as :

[tex]F_N=mg\ cos\theta[/tex]

[tex]F_N=850\times \ cos20[/tex]

[tex]F_N=798.73\ N[/tex]

So, the normal force of the roof on the worker is 798.73 N. Hence, this is the required solution.

The magnitude of the normal force of the roof worker is 798.66 N

The parameters given in the question are

Weight= 850 N

angle=  20°

The formula for calculating the magnitude of the normal force of the roof worker is

= mg cos 20°

= 850 × 0.9396

= 798.66

Hence the magnitude of the normal force of the roof worker is 798.66 N

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Answer:

At 15cm to the left of the pivot

Explanation:

The weight 0.15kg(1.5N) and 3.2N are balanced on a meter stick. This weights are parallel to each other on the stick. Questions on parallel forces acting on a body are solved using the *principle of moment."

Moment is the turning effect of force about a point.

Principle of moment states that the sum of clockwise moment is equal to the sum of anticlockwise moments.

According to the question, the meter (100cm) stick is balanced with a pivot @ 18cm mark {this is the shorter end}

If the 3.2N weight is hung at xcm from the shorter end, this weight will turn about the pivot in the anticlockwise direction.

Since moment = Force × perpendicular distance from the force

Moment of the 3.2N weight = 3.2N × x = 3.2x

The weight of the stick (1.5N) will be positioned at the center of the stick i.e @ the 50cm mark which is at 32cm to the right of the pivot.

The weight of the stick will turn in the clock wise direction

The moment of the weight in the clockwise direction = 1.5×(50-18) = 1.5×32 = 48Ncm

Using the principle of moment, we will equate both moments to have;

3.2x = 48

x = 48/3.2

x = 15cm

This means the 3.2N weight will be positioned at the 15cm mark to the left if the pivot to balance the meter stick.

To find the force that the boy is exerting on the stuffed animal, subtract the net force from the girl's force. The boy is exerting a force of 3N.

To find the force that the boy is exerting, we first need to determine the net force acting on the stuffed animal. The net force is equal to the mass of the stuffed animal multiplied by its acceleration. In this case, the mass is 0.2 kg and the acceleration is 2.5 m/s^2. So the net force is 0.2 kg * 2.5 m/s^2 = 0.5 N.

Since the girl exerts a force of 3.5 N, and the net force is 0.5 N, the boy's force must be equal to the difference between these two forces. Therefore, the boy is exerting a force of 3.5 N - 0.5 N = 3 N.

The correct statements are:

B. a small rock sitting on top of a big rock

As the rock is at a height with respect to ground it has potential Energy

and

C. a stretched rubber band

A stretched rubber band has elastic potential energy

The others are actually moving and hence would consist of Kinetic energy. Potential energy is stored in objects that do not move and are stationary.

Using basic kinematics and trigonometry in projectile motion, we find the horizontal and vertical velocity components, time to reach the highest point, and the maximum height. We then determine the range of the projectile and the acceleration and velocity components at the highest point.

Let's solve the question step by step:

(a) Horizontal and Vertical Components of Velocity

To find the components of the shell's initial velocity, we can use trigonometry:

• Horizontal component (vx): vx = v * cos(θ) = 70.0 m/s * cos(60.0°)

• Vertical component (vy): vy = v * sin(θ) = 70.0 m/s * sin(60.0°)

(b) Time to Reach the Highest Point

The time to reach the highest point can be determined using the formula: t = vy / g, where g is the acceleration due to gravity (9.8 m/s2).

(c) Maximum Height

The maximum height is given by H = (vy2) / (2g).

(d) Horizontal Range

The horizontal range can be calculated once we have the total time of flight and the horizontal velocity: range = vx * total time of flight.

(e) Components of Acceleration and Velocity at the Highest Point

At the highest point:

• Horizontal velocity component remains unchanged: same as initial vx.

• Vertical velocity component is 0 m/s because the projectile stops rising and is about to start falling.

• Horizontal acceleration is 0 m/s2 because no forces are acting in the horizontal direction.

• Vertical acceleration remains -9.8 m/s2 (negative because it is directed downwards).

By definition we have to:

A system or frame of reference are those conventions used by an observer (usually standing at a point on the ground) to be able to measure the position and other physical magnitudes such as speed and acceleration of one or several objects.

The numerical value of some magnitudes can also be relative to the reference system when we refer to the relative movement. There are always mathematical relationships between the observer and the relative magnitudes.

Answer:

C. a position from which something is observed

All waves, regardless of their type, carry energy from one place to another without transporting matter. They do not all carry light, matter, or particles.

All waves, regardless of whether they are light waves, sound waves, electromagnetic waves, or even water waves, carry energy. Therefore, the answer to your question is a) energy.

Waves transfer energy from one place to another without transporting matter. They do this through oscillations and disturbances in a medium. For example, when you throw a stone into a pond, the waves created by the stone do not transport the water from one side of the pond to another. Instead, they transport the energy created by the stone's impact.

Not all waves carry light, matter, or particles. Light is specifically carried by electromagnetic waves, while particles and matter are not typically associated with waves in the conventional sense.

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Final answer:

The mass attached to the elastic band travels approximately 12.04 centimeters from its equilibrium position, as calculated by the amplitude of the given motion equation.

Explanation:

To determine how far from its equilibrium position the mass travels, we must find the amplitude of the motion described by the given equation: s = 8 cos t + 9 sin t. The amplitude can be found by recognizing that the equation represents the sum of two simple harmonic motions, and we use the Pythagorean theorem to find the resultant amplitude.

The general formula for the amplitude, A, when given a cos t + b sin t is A = sqrt(a^2 + b^2). Substituting the given coefficients 8 and 9 into this formula gives us the amplitude A = sqrt(8^2 + 9^2) = sqrt(64 + 81) = sqrt(145), which is approximately A ≈ 12.04 cm.

Thus, the mass travels approximately 12.04 centimeters from its equilibrium position.