Dynamics

Dynamics II: Motion in a Plane

Dynamics in Two Dimensions

Before, we always had one of ax or ay being zero. 1-D dynamics

But there are lots of cases where the motion is not along a line.

ConcepTest 1

1. A

2. B

3. C

4. D

65%

3%

29%

3%

1

2

3

4

A hockey puck is pushed across frictionless ice in the direction shown. The puck receives a sharp, very short duration kick toward the right as it crosses line 2. It receives a second kick, of equal strength and duration but toward the left, as it crosses line 3. What diagram best represents the motion of the puck?

ConcepTest 2

1. A

2. B

3. C

4. D

1%

2%

90%

8%

1

2

3

4

A rocket motor is taped to an ice hockey puck. The puck is given a push in the direction shown; the rocket is turned on by remote control as the puck crosses line 2, then turned off as it crosses line 3. Which diagram best represents the motion of the puck?

Example Rocketing in the Wind

1. A

2. B

3. C

46%

34%

19%

A small rocket for gathering weather data has a mass of 30 kg and generates 1500 N of thrust. On a windy day, the wind exerts a 20 N horizontal force on the rocket. If the rocket is launched straight up, what is the shape of the trajectory? (Assume no mass loss.) First, guess:

Uniform Circular Motion Revisited

Recall that a particle in uniform circular motion has a centripetal acceleration

Lets use a new coordinate system:

Uniform Circular Motion Revisited

Then the velocity of a particle in uniform circular motion can be written

and the acceleration will be

Sliding in a Horizontal Circle

1. Up

2. Down

3. Toward the centre of the circle

4. Away from the centre of the circle

5. There is no net force on the keys

0%

0%

0%

0%

0%

Consider my keys sliding on the table in a circle. Assume theres no friction. Draw a free body diagram for the keys using the rtz coordinate system. Where does the net force point?

Swinging in a Horizontal Circle

1. Along the string towards the centre of the circle

2. Along the string away from the centre of the circle

3. Horizontally towards the centre of the circle

4. Horizontally away from the centre of the circle

5. There is no net force on the keys

53%

14%

32%

2%

0%

Now my keys will be in the air. Same question: draw a free body diagram. Where does the net force point?

Dynamics of Uniform Circular Motion

A particle in uniform circular motion is undergoing acceleration, so there must be a net force acting on it.

How much net force does it need to travel in a circle?

Newtons 2nd law tells us

Example: Turning the Corner

A concrete highway curve of radius 70 m is banked at a 15d angle. What is the maximum speed with which a 1500 kg rubber-tired car can take this curve without sliding?

Circular Orbits

Weve been treating the planet as a flat surface, but of course it isnt:

Circular Orbits

So, an orbiting object is in free fall. The only force on it is gravity.

This is why astronauts are weightless in orbit.

The Water Bucket Experiment

Imagine swinging a bucket of water over your head. If you swing it quickly, the water will stay in the bucket.

If you swing it slowly, the water will fall out.

Why?

Never mind for now. Lets think about roller coasters instead.

Roller Coasters

1. Yes

2. No

3. Not enough information to tell

0%

0%

0%

Consider a simple roller coaster shown below. Is the cart, as it travels the circle, in uniform circular motion?

Roller Coasters

Draw free body diagrams for the cart at the bottom, right, top, and left positions.

Roller Coasters

Lets analyze the free body diagram for the cart at the bottom of the track:

Roller Coasters

Lets analyze the free body diagram for the cart at the top of the track:

Roller Coasters

What happens if vtop is not very large? What if n = 0?

At what speed will the normal force be zero?

Back to the Bucket

The reason the water stays in the bucket is exactly the same as for the roller coaster.

We can write the critical velocity in terms of the angular velocity:

ConcepTest 3

1. N > FG 2. N < FG 3. N = FG

4. We cant tell n without knowing v.

0%

0%

0%

0%

An out-of-gas car is rolling over the top of a hill at speed v. At this instant,

ConcepTest 4

1. (a)

2. (b)

3. (c)

4. (d)

0%

0%

0%

0%

A ball on a string is swung in a vertical circle. The string happens to break when it is parallel to the ground and the ball is moving up. Which trajectory does it follow?

Final thoughts

That ends forces! But its a big subject, and doing all the practice questions is highly recommended.