**Ch7 - Circular Motion and Gravitation - Notes **
1 – Centripetal Acceleration and Force

**This unit we will investigate the special case of kinematics and dynamics of objects in uniform circular motion. **

First let’s consider a mass on a string being twirled in a horizontal circle at a constant speed.

Ok so we’ve figured out its speed, but is the mass accelerating?

Remember that the mass is traveling at a constant speed. However, acceleration is defined as:

**So how does the velocity of the mass change with respect to time? **

Notice that the direction of the velocity at any time is …

So even though it may be traveling at a _________ _________ anything traveling in a circular path is
**_________________ because the _________________ of its velocity is always changing. **

Let’s determine the speed of the object.

Remember that speed is defined as:

We define the period of motion (T) as the time it takes to complete one rotation.

How far does it travel in one rotation?

We can find the circumference of the circular path (distance traveled) by:

Therefore the speed of an object in uniform circular motion is:

The acceleration of an object in uniform circular motion is:

**T is Period: **

Let’s do a quick derivation of this formula:

It is worth noting from the above

derivation that the direction of the change in velocity is always....

Therefore the acceleration of an object in circular motion is always towards the…

**This is the definition of centripetal, **
**which means center-seeking. **

Whenever an object is accelerated there must be a…

**This force is known as centripetal force, F**c**. This is not a new force, it is simply the net force that **
accelerates an object towards the center of its circular path.

Examples:

1) A mass is twirled in a circle at the end of a string, the centripetal force is provided by _______________________________

2) When a car rounds a corner on a highway, the centripetal force is provided by _______________________________

3) When the Moon orbits the Earth, the centripetal force is provided by _______________________________

Newton’s Second Law we can help us to determine a formula for centripetal force:

Example:

A 0.50 kg mass sits on a frictionless table and is attached to hanging weight. The 0.50 kg mass is whirled in a circle of radius 0.20 m at 2.3 m/s.

Calculate the centripetal force acting on the mass.

Calculate the mass of the hanging weight.

Example:

A car traveling at 14 m/s goes around an unbanked curve in the road that has a radius of 96 m. What is its centripetal acceleration?

What is the minimum coefficient of friction between the road and the car’s tires?

Example:

A plane makes a complete circle with a radius of 3622 m in 2.10 min. What is the speed of the plane?

**One last note on a little thing called centrifugal force. While centripetal means center-___________________ **

centrifugal means center- _______________________.

**An inertial frame of reference is a one where Newton’s Law’s ________ ____________. **

*In an inertial frame of reference, centrifugal force is actually an apparent force - it does not exist. It is simply the *
apparent force that causes a revolving or rotating object to move in a straight line.

However, Newton’s First Law tells us that we do not need a force to keep an object moving in a straight line, you
*only need a force to deflect an object from moving in a straight line. *

Example:

When riding in the backseat of a car that is turning a corner, you slide across the seat, seeming to accelerate outwards, away from the center of the turning circle.

Explain why the force in this case is actually working towards the center of the turn and not outwards.

Practice Problems – Ch 7-1 Centripetal Acceleration & Force

1) Calculate the centripetal force acting on a 925 kg car as it rounds an unbanked curve with a radius of 75 m at a speed of 22 m/s.

2) A small plane makes a complete circle with a radius of 3282 m in 2.0 min. What is the centripetal acceleration of the plane?

3) A car with a mass of 833 kg rounds an unbanked curve in the road at a speed of 28.0 m/s. If the radius of the curve is 105 m, what is the average centripetal force exerted on the car?

4) An amusement park ride has a radius of 2.8 m. If the time of one revolution of a rider is 0.98 s, what is the speed of the rider?

5) An electron (m=9.11x10^{-31} kg) moves in a circle whose radius is 2.00 x 10^{-2} m. If the force
acting on the electron is 4.60x10^{-14} N, what is its speed?

6) A 925 kg car rounds an unbanked curve at a speed of 25 m/s. If the radius of the curve is 72 m, what is the minimum coefficient of friction between the car and the road required so that the car does not skid?

7) A 2.7x10^{3} kg satellite orbits the Earth at a distance of 1.8x10^{7} m from the Earth’s centre at a
speed of 4.7x10^{3} m/s. What force does the Earth exert on the satellite?

8) A string can withstand a force of 135 N before breaking. A 2.0 kg mass is tied to the string and whirled in a horizontal circle with a radius of 1.10 m. What is the maximum speed that the mass can be whirled at before the string breaks?

9) A 932 kg car is traveling around an unbanked turn with a radius of 82 m. What is the maximum speed that this car can round this curve before skidding:

a) if the coefficient of friction is 0.95?

b) if the coefficient of friction is 0.40?

1.) 6.0x10^{3} N 2.) 9.0 m/s^{2} 3.) 6.2x10^{3} N 4.) 18 m/s 5.) 3.18x10^{7} m/s
6.) 0.89 7.) 3.3x10^{3} N 8.) 8.62 m/s 9.a.) 28 m/s b.) 18 m/s

**Circular Motion and Gravitation Notes (Ch 7.3 – Gravitation) **
**Artificial Gravity **

Example 1:

* If this spinning cylindrical spacecraft has a radius of 13 meters (about the length of this room), how fast *
does it have to spin to simulate earth gravity? What about ¼ of earth g?

Example 2:

In the game Halos, the Halo Superstructure has a radius of 8,000 km. How fast would the Halo Superstructure have to spin to simulate earth gravity?

Newton discovered that gravity attracts any two objects depending on their

____________________ and their ______________________.

Fg is proportional to the product of the two masses, and Fg is inversely proportional to the square of the distance between their centers of mass

Example

Calculate the force of gravity between two 75 kg students if their centers of mass are 0.95 m apart.

Example

What is the gravitational force if the mass of one of the students in the previous problem is doubled?

What is the gravitational force if distance between them is halved?

While we’re at it, let’s make sure we clear up another common misconception: mass vs. weight.

Mass: Weight:

Where:

Newton’s Law of Universal Gravitation

We were already familiar with the force of gravity on
earth. Is it the same as Newton’s universal gravity
equation? mEarth = 5.97x10^{24} kg rEarth = 6.37x10^{6} m

Acceleration of Gravity

Where:

Calculate “g” (the acceleration of gravity) on Mars
(mMars = 6.42x10^{23} kg rMars = 3.37x10^{6} m)

How much would a 70 kg person weigh on Earth?

How much would they weigh on Mars?

Philae (the comet lander) has a mass of 100kg. The comet it landed on (67P/Churyumov–Gerasimenko)
has a mass of 1.0 x 10^{13} kg and a radius of approximately 1.7 km.

1. What was Philae’s weight on earth?

2. What was its weight on the comet?

Practice Problems – Ch 7-2 Artificial Gravity & Newton’s Universal Gravity Equation

1. A spinning cylindrical spacecraft has a radius of 25 meters.

How fast does it have to spin to simulate earth gravity?

5. What gravitational force does the moon produce
on the Earth if their centers are 3.84x10^{8} m apart?

(mEarth = 5.98x10^{24} kg mMoon = 7.35x10^{22} kg)

2. A cylindrical spacecraft can spin with a tangential velocity (v) of 14 m/s. How large is the radius if it will simulate earth gravity?

6. If the gravitational force between two objects of
equal mass is 2.30x10^{-8} N when the objects are 10.0
m apart what is the mass of each object?

3. How much artificial gravity do you feel on a cylindrical spacecraft spinning at 0.35 rad/s with a radius of 55 m?

7. What is the acceleration of gravity on the moon?

(rmoon = 1.74x10^{6} m)

4. Two students are sitting 1.50 m apart. One student has a mass of 70.0 kg and the other has a mass of 52.0 kg. What is the gravitational force between them?

8. How much does a 70 kg person weigh on earth?

How much do they weigh on the moon?

1. 16 m/s or 0.63 rad/s 2. 20 m 3. 6.7 m/s^{2} or 0.69g 4. 1.08x10^{-7} N 5. 1.99x10^{20} N
6. 186 kg 7. 1.62 m/s2 8. Earth: 686 N moon: 113 N

**Ch 7: Gravity Continued: Satellite Orbits **

Bellringer: Cere’s is a dwarf planet (like Pluto, but way closer to us). It hangs out between Mars and Jupiter
in the Asteroid belt. Ceres has a radius of 476.2 km and a mass of 9.43 x 10^{20} kg. If a person weighs 120
pounds on Earth, how much would they weigh on Ceres?

(1 kg = 2.20 lbs on Earth).

Draw the path a satellite takes when it orbits earth, and label the force(s) on the satellite, label altitude, rplanet, rorbit:

NOTE: Altitude =

A satellite of the Earth, such as the moon, is constantly falling. But it does not fall towards the Earth, rather it falls around the Earth. It’s kind of like if you were in an elevator that was falling towards the Earth, you would feel weightless. Satellites are falling around the Earth, and feel weightless. Gravity and centripetal motion keep them going in circles though.

The only force acting on a satellite in
circular orbit is gravity, so… gravity
**is the centripetal force! **

1. Start with: F_{c} = F_{g}

or

2. Solve for unknown 3. Plug in known values

Example: The International Space Station has a mass of 450,000 kg and orbits the Earth at an average altitude of about 357 km.

How fast is it going?

How long does it take to go around the earth once?

**MOON: The Earth and the Moon’s centers are 3.84 × 10**^{8}m apart. How long does it take the Moon
to make a full trip around the Earth? 𝑚_{𝑒𝑎𝑟𝑡ℎ} = 5.98 × 10^{24}kg 𝑚_{𝑚𝑜𝑜𝑛} = 7.35 × 10^{22}kg

**DAWN: The Dawn Spacecraft is set to rendezvous with Ceres on March 6**^{th}. It will be the first

mission to study a dwarf planet at close range!!. Dawn will study Ceres at 4 different orbits. Ceres has
a radius of 476 km and m = 9.43 × 10^{20}𝑘𝑔. Calculate the missing info to help NASA determine the
necessary velocity and resulting period at which they must orbit.

**Date ** **Altitude **

**(not **
**radius) **

**r****c**** **
**(orbital **
**radius) **

**Velocity (∆𝒗 is **
**achieved using 3 **
**xenon ion **

**thrusters) **

**Period of **
**orbit (T) **
**(hrs or **
**days) **
March 6, 2015

(first full

characterization)

13500 km

June 2015 (survey orbit)

4430 km August 2015

(high-altitude mapping orbit)

1480 km

November 2015 (close orbit, future moon)

375 km

Practice Problems – Ch 7-3 Orbital Motion

1. Calculate the gravitational force on a 6.50x10^{2} kg
spacecraft that is 4.15x10^{6} m above the surface of the

Earth.

2. A satellite in orbit around the earth has a velocity of 3066 m/s. What is its orbital radius? How long does it take to make one full trip around the earth?

3. Mercury, the closest planet to the sun, has an orbital
radius of about 5.79x10^{10}m. How long (in Earth days) is
a year on Mercury (1 year means 1 full trip around the
sun).

(mSun = 1.99x10^{30} kg mMercury = 3.3x10^{23} kg)

4. If the earth orbited in a perfect circle (it’s actually slightly elliptical), how far must it be from the sun since our year is 365.24 days?

(mSun = 1.99x10^{30} kg mEarth = 5.98x10^{24} kg)

1. (2.34x103 N) 2. (r=4.243x107m, T= about 1 day) 3. (88 days) 4. (1.5x10^{11}m)

**Aerospace Engineer for a Day: Project **

You just landed your dream job at NASA and get to make a satellite that will explore any planet or dwarf planet in our solar system. You must pick a planet to study and plan out your mission.

**1. Pick any planet or dwarf planet in our solar system (must be different from your **

neighbors’).

### ______________________________

**2. Search online and find: **

**a. Mass of the object: ** **_____________________________ **

**b. Radius of the object: ** **_____________________________ **

**c. Avg. distance from Sun: _____________________________ **

**d. Length of 1 year (1 trip around the sun) in earth days: ** **___________________ **

**i. Convert this time to seconds (show work below): ___________________ **

**3. Using Newton’s universal gravitation equation, determine the acceleration of gravity on **
**the planet (show work). **

**4. Using the universal gravitation equation combined with our equations for centripetal **
**motion, answer the following questions: **

**a. Calculate the speed the planet is traveling around the sun (using only the sun’s **
**mass and the average distance from the sun): **

**b. Calculate the orbital period (amount of time for 1 trip around the sun), (using only **
the sun’s mass and the average distance from the sun), and compare this value
**from what you found on-line: **

**5. You are going to send a spacecraft to your planet to study it from a distance. You want to **
put the spacecraft in orbit around the planet. First, you want to study it from afar, then
move in closer and stay there for a while, then move in closer again, and finally, you
want to get in really close. Fill in the following table to calculate where your satellite
**needs to be in orbit and how fast it must go to maintain that orbital radius. **

**Orbit Type ** **Altitude **
**(not **
**radius) **

**r****c**** **
**(orbital **
**radius) **

**Velocity (∆𝒗 is **
**achieved using 3 **
**xenon ion **

**thrusters) **

**Period of **
**orbit (T) **
**(hrs or **
**days) **
First full

characterization

13500 km

Survey Orbit 4430 km High-altitude

mapping orbit

1480 km

Close orbit (future moon)

375 km

**Show work below for the first orbit (“First full characterization”): **