## A particle of mass m rotates with a uniform speed on the inside of a bowl’s parabolic frictionless surface in a horizontal circle of radius R = 0.4 meters as shown below. At the position of the particle the surface makes an angle θ = 17o with the vertical. What is the angular velocity of the particle? ## A particle of mass m rotates with a uniform speed on the inside of a bowl’s parabolic frictionless surface in a horizontal circle of radius R = 0.4 meters as shown below. At the position of the particle the surface makes an angle θ = 17o with the vertical. What is the angular velocity of the particle?
## To study Newton’s Law of Gravitation ## To consider gravitational force, weight, and gravitational energy ## To compare and understand the orbits of satellites and celestial objects
**Every particle** of matter in the universe attracts every other particle with a force that is **directly **proportional to the product of the **masses** of the particles and **inversely** proportional to the **square of the distance** between them. **Every particle** of matter in the universe attracts every other particle with a force that is **directly **proportional to the product of the **masses** of the particles and **inversely** proportional to the **square of the distance** between them.
## Lowercase **g** is the **acceleration due to gravity**, which relates the weight **w** of a body to its mass **m: w = mg**. The value of **g** is **different at different locations** on the earth's surface and on the surfaces of **different planets.** ## Lowercase **g** is the **acceleration due to gravity**, which relates the weight **w** of a body to its mass **m: w = mg**. The value of **g** is **different at different locations** on the earth's surface and on the surfaces of **different planets.** ## By contrast, capital **G **relates the gravitational force between any two bodies to their masses and the distance between them. Henry Cavendish measured the value of G in 1798. We call G a universal **constant** because it has the same value for any tow bodies, no matter where in space they are located.
## We defined the weight of a body as the attractive gravitational force exerted on it **by the earth**. ## We defined the weight of a body as the attractive gravitational force exerted on it **by the earth**. ## The broaden definition of weight is: The weight of a body is the total **gravitational force** exerted on the body by **all other bodies** in the universe. ## When the body is near the surface of **the earth**, we can neglect all other gravitational forces and consider the weight as just the earth’s gravitational attraction. ## At the surface of the **moon **we consider a body’s weight to be the gravitational attraction of the moon, so on.
## If we model the earth as a spherically symmetric body with radius **RE** and mass **mE**, the weight **w**** **of a small body of mass m at the earth’s surface is ## If we model the earth as a spherically symmetric body with radius **RE** and mass **mE**, the weight **w**** **of a small body of mass m at the earth’s surface is
## Suppose we drill a hole through the earth (radius *RE*, mass *mE*) along a diameter and drop a mail pouch (mass *m*) down the hole. Derive an expression for the gravitational force on the pouch as a function of its distance* r *from the center. Assume that the density of the earth is uniform. ## Suppose we drill a hole through the earth (radius *RE*, mass *mE*) along a diameter and drop a mail pouch (mass *m*) down the hole. Derive an expression for the gravitational force on the pouch as a function of its distance* r *from the center. Assume that the density of the earth is uniform.
## Gravity (and hence, weight) decreases as altitude rises outside Earth and also decrease as approaching the center of Earth and becomes zero at the center of Earth. ## Gravity (and hence, weight) decreases as altitude rises outside Earth and also decrease as approaching the center of Earth and becomes zero at the center of Earth.
## The apparent weight of a body on earth **differs** slightly from the earth’s gravitational force because the earth rotates and is therefore **not** precisely an **inertial frame of reference**. ## The apparent weight of a body on earth **differs** slightly from the earth’s gravitational force because the earth rotates and is therefore **not** precisely an **inertial frame of reference**. ## We have used the fact that the earth is an approximately spherically symmetric distribution of mass. But this does **not** mean that the earth is uniform. In fact, the density of the earth decreases with increasing distance from its center.
## Rank the following hypothetical planets in order from highest to lowest surface gravity: ## Rank the following hypothetical planets in order from highest to lowest surface gravity: ## Mass = 4 times the mass of the earth, radius = 4 times the radius of the earth ## Mass = 4 times the mass of the earth, radius = 2 times the radius of the earth; ## Mass = 2 times the mass of the earth, radius = 4 times the radius of the earth.
## Ball *B*** (**moment of inertial about its center ⅔ MR2), rolls down the distance* x *along the inclined plane without slipping. In terms of acceleration due to earth’s gravity *g*, what is the acceleration of ball *B* along the inclined plane? ## Ball *B*** (**moment of inertial about its center ⅔ MR2), rolls down the distance* x *along the inclined plane without slipping. In terms of acceleration due to earth’s gravity *g*, what is the acceleration of ball *B* along the inclined plane?
## MC packet: #21, 23, 24, 25, 32,36 ## MC packet: #21, 23, 24, 25, 32,36
## To determine the gravitational potential energy of an object at a height that is way above the earth’s surface, we need to calculate work done by the gravitational force on the object and use equation: ## To determine the gravitational potential energy of an object at a height that is way above the earth’s surface, we need to calculate work done by the gravitational force on the object and use equation:
## Gravitational potential energy depends on the distance **r** between the body of mass **m** and the center of the earth. When the body moves away from the earth, **r** increases, the gravitational force does negative work, and **U** increases (becomes less negative). When the body “falls” toward earth, **r** decreases, the gravitational work is positive, energy decreases (becomes more negative). ## Gravitational potential energy depends on the distance **r** between the body of mass **m** and the center of the earth. When the body moves away from the earth, **r** increases, the gravitational force does negative work, and **U** increases (becomes less negative). When the body “falls” toward earth, **r** decreases, the gravitational work is positive, energy decreases (becomes more negative). ## Note that U is **relative**. Only **∆U** is significant.
## Extra credit on test – use the above equation to prove that near Earth’s surface the potential energy is ## Extra credit on test – use the above equation to prove that near Earth’s surface the potential energy is
## 1st Law: Each planet moves in an elliptical orbit with the sun at one focus. ## 1st Law: Each planet moves in an elliptical orbit with the sun at one focus. ## 2nd Law: A line connecting the sun to a given planet sweeps out equal areas in equal times.
**Perihelion** is the point closest to the sun; **Perihelion** is the point closest to the sun;
**Aphelion** is the point furthest from the sun.
**Semi-Major axis **is the length of *a*
**ea = e∙a: **the distance from each focus to the center of the ellipse.
**Eccentricity (e): **a dimensionless number between 0 and 1. If e=0, ea=0, the ellipse is a circle.
## The orbit of Comet X has a semi-major axis that is four times larger than the semi-major axis of Comet Y . What is the ratio of the orbital period of X to the orbital period of Y? ## The orbit of Comet X has a semi-major axis that is four times larger than the semi-major axis of Comet Y . What is the ratio of the orbital period of X to the orbital period of Y?
## The planet Saturn has about 100 times the mass of the earth and is about 10 times farther from the sun than the earth is. Compared to the acceleration of the earth caused by the sun’s gravitational pull, how great is the acceleration of Saturn due to the sun’s gravitation? [show work] ## The planet Saturn has about 100 times the mass of the earth and is about 10 times farther from the sun than the earth is. Compared to the acceleration of the earth caused by the sun’s gravitational pull, how great is the acceleration of Saturn due to the sun’s gravitation? [show work] ## 10 times greater ## The same ## 1/10 as great ## 1/100 as great.
## How far from a very small 150 kg ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does? ## How far from a very small 150 kg ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does?
## What is the escape speed from an asteroid of diameter 255 km with a density of 2720 kg/m3 ? ## What is the escape speed from an asteroid of diameter 255 km with a density of 2720 kg/m3 ?
## How far from a very small 120 kg ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does? ## How far from a very small 120 kg ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does? ## Is it reasonable that you could actually set up this as an experiment? Why?
## On Earth’s surface, r1 ≈ r2 = RE; (radius of the earth) and ## On Earth’s surface, r1 ≈ r2 = RE; (radius of the earth) and ## r2 – r1 = h (height above Earth’s surface)
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