If an object which weighs 100 lbs on the Earth's surface were placed on a planet with 2 times the radius of the Earth and with 7 times the Earth's mass, how much would that object weigh? Enter answer to nearest 0.1 lbs.
The term "void ratio" means the ratio of the volume of empty space to volume of occupied space. It is a measure of how empty a volume is. Within stellar systems, the average separation between planets is roughly 10 to the 15 cm and the average planetary radius is typically 10 to the 15 cm. The void ratio for this stellar system is roughly 10 to what power: [Hint: Remember that volume is proportional to the cube of the linear size]
Given that the acceleration of gravity at the Earth's surface is about 980 cm per sec per sec, the centripetal acceleration [v squared over r] of an artificial satellite in a circular orbit with a radius 3.9 times that of the Earth's radius would be about _____ cm per sec per sec. (Hint: F = m a, and the force of gravity gets weaker as the square of the distance, so how much weaker is gravity at 3.9 times the Earth's Radius than it is at the surface of the Earth?)
If the distance between the Earth and the Sun were increased by a factor of 4.56, by what factor would the strength of the force between them change? [Hint: Use Newton's Law of Universal Gravitation, and give your answer to 2 decimal places only]
If a certain force accelerates an object of mass 27 Kg at 50 m/s/s, what acceleration in m/s/s would the same force produce on another object of mass 27? Enter answer to at least one decimal place to the right of the decimal point.
Show work please I need to know how to do it
#1 You know that the weight is
GMm/r^2 = 100
Replace M by 7M and r by 2r, and you have the new weight of
G(7M)m/(2r)^2 = 7/4 GMm/r^2 = 175
#2
Huh? The planetary radius is equal to the separation? Do the planets touch, or is the separation between planetary surfaces?
If so, then a cube with side 5r contains about 9 planets of radius r. So, the void ratio would be
((5r)^3 - 9*4/3 π*r^3)/(9*4/3 π*r^3)
= (125-12π)/12π
≈ 2.32
To calculate the weight of an object on a different planet, we can use the formula:
Weight on new planet = (mass on new planet) * (acceleration due to gravity on new planet)
1. First, let's find the mass on the new planet. We know that the mass on Earth is 100 lbs. Since the planet has 7 times the Earth's mass, the mass on the new planet would be:
Mass on new planet = 7 * 100 lbs = 700 lbs
2. Next, we need to find the acceleration due to gravity on the new planet. We know that the radius of the new planet is 2 times the Earth's radius. Since the acceleration due to gravity is inversely proportional to the square of the radius, we can use the following formula:
Acceleration due to gravity on new planet = (acceleration due to gravity on Earth) * (Earth's radius / new planet's radius)^2
Acceleration due to gravity on Earth is approximately 980 cm/s^2. Let's convert it to a more suitable unit, such as ft/s^2:
Acceleration due to gravity on Earth = 980 cm/s^2 * (1 ft / 30.48 cm)^2 = 32.17 ft/s^2 (approximately)
Now, let's substitute the values into the formula:
Acceleration due to gravity on new planet = 32.17 ft/s^2 * (1 ft / (2 ft))^2
= 32.17 ft/s^2 * (1 / 4)
= 8.04 ft/s^2
3. Finally, we can calculate the weight on the new planet using the formula mentioned earlier:
Weight on new planet = Mass on new planet * Acceleration due to gravity on new planet
= 700 lbs * 8.04 ft/s^2
= 5628 lbs (approximately)
Therefore, the object would weigh approximately 5628 lbs on the new planet.
Regarding the second question:
The void ratio is the ratio of the volume of empty space to the volume of occupied space. In this case, the void ratio is roughly 10^15 cm / 10^15 cm, which simplifies to 1.
As for the third question:
The centripetal acceleration of an artificial satellite in a circular orbit can be calculated using the formula:
Centripetal acceleration = (velocity squared) / radius
The force of gravity gets weaker as the square of the distance, so the acceleration at 3.9 times the Earth's radius would be 1/((3.9)^2) = 1/15.21 times the acceleration at the Earth's surface.
Let's substitute the values into the formula:
Centripetal acceleration = (acceleration due to gravity on Earth) / [(3.9)^2]
= 980 cm/s^2 / (3.9)^2
= 980 cm/s^2 / 15.21
= 64.26 cm/s^2 (approximately)
Therefore, the centripetal acceleration of an artificial satellite in a circular orbit with a radius 3.9 times that of the Earth's radius would be approximately 64.26 cm/s^2.
Lastly, for the fourth question:
If the distance between the Earth and the Sun is increased by a factor of 4.56, the strength of the force between them would change by the inverse square of the distance.
Let's calculate the change in force:
Change in force = (new distance / old distance)^2
= (4.56)^2
= 20.7936
Therefore, the strength of the force between the Earth and the Sun would change by a factor of approximately 20.79.
For the last question:
If a certain force accelerates an object of mass 27 kg at 50 m/s^2, the same force would produce the same acceleration on another object of mass 27 kg.
Therefore, the acceleration would be 50 m/s^2.
I hope this helps! Let me know if you have any further questions.
To answer these questions, we need to use various equations related to force, gravity, and mass. Here's how you can approach each question:
1. Weight on a different planet:
The weight of an object is given by the formula: weight = mass * acceleration due to gravity. In this case, the mass remains the same at 100 lbs. However, the acceleration due to gravity on the new planet would be different. We can calculate it using the equation: acceleration due to gravity = (gravitational constant * mass of the planet) / (radius of the planet)^2. Plugging in the values, we find the acceleration due to gravity on the new planet. Finally, we can calculate the weight on the new planet using the same weight formula.
2. Void ratio:
The void ratio is given by the formula: void ratio = empty space volume / occupied space volume. Here, we are given the average separation between planets and the average planetary radius. We need to calculate the volumes of both empty and occupied space, and then divide them to get the void ratio. Remember that volume is proportional to the cube of the linear size, so we need to consider the cubic powers of the distances given.
3. Centripetal acceleration of a satellite:
The formula for centripetal acceleration is given by: centripetal acceleration = (velocity)^2 / radius. In this case, we are given the radius of the artificial satellite's circular orbit but not its velocity. However, we can infer the velocity using the fact that the gravitational force gets weaker with the square of the distance. We can calculate the force of gravity at the new distance, and then use the formula for gravitational force to find the velocity. Finally, we can use this velocity in the centripetal acceleration formula.
4. Change in force between the Earth and the Sun:
According to Newton's Law of Universal Gravitation, the force between two objects is given by the formula: force = (gravitational constant * mass1 * mass2) / (distance between them)^2. Here, we are given the factor by which the distance between the Earth and the Sun is increased. We can calculate the new distance, and then use the force formula to find the change in force. Divide the new force by the original force to get the factor by which it changes.
5. Acceleration on another object:
If the same force is applied to two objects of different masses, the acceleration will be inversely proportional to the mass. So, we can use the equation: acceleration1 / acceleration2 = mass2 / mass1. Given the mass and acceleration of the first object, we can substitute the values and solve for the acceleration of the second object.
By following these steps and using the appropriate formulas, you can find the answers to each question.