Motion

Inertia

A body in motion tends to remain in motion; a body in rest tends to remain at rest.

How to Describe Motion?

• Position: Where you are
• Velocity: Consists of two parts:
  1. How fast you are going (speed)
  2. Which way you are going (direction)
• Acceleration: How fast your velocity is changing:
  • Speeding up
  • Slowing down
  • Changing direction

Newton's Law

• First Law: An object that is not subject to any outside forces moves at constant velocity, covering equal distances in equal times along a straight line path.
• Second Law: The force exerted on an object is equal to the product of the object's mass times its acceleration. The acceleration is in the same direction of the force.
  • \(F=ma\)
  • Easier to move a light object than a heavy one
• Third Law: For every force that one object exerts on a second object, there is an equal but oppositely directed force that the second objects exerts on the first object.

Gravity

Left Picture:

• Need to consider all the forces acting on the body.
• Normal force from ground pushes up on stick man
• \(F_{Total} = N - F = 0\)

Right Picture:

• \(F_{Total} = -F\)
• Accelerate down

Falling ball

\(F = mg = ma\)

\(g=a\)

Acceleration independent of mass

Note

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ChatGPT said:

In a vacuum, where there is no air resistance, both a rock and a feather would drop at the same rate and hit the ground at the same time. So, feather falls as fast as a lead brick.

Energy

Left Picture:

Higher has more potential energy

Right Picture:

Faster moving has more kinetic energy

Work

• Work = Force \(\times\) distance
• Work = \(mgh\)
• If you are not pushing or it is not moving, then you are not working.

Momentum

• Tendency to continue moving in a certain direction
• More momentum: Tend to win in collisions
• Momentum Equation: Momentum = Mass x Velocity
  • more mass = more momentum
  • more velocity = more momentum

Pushing a block

• \(F_{gravity} = N\) (block not falling)
• \(F_{push} = ma\) (the block accelerates)

Friction

Friction slows the motion.

• \(F_{gravity} = N\) (block not falling)
• \(F_{push}-F_{friction} = ma\) (the block accelerates)
• \(F_{push}\) needs to be greater than \(F_{friction}\) for motion to occur.
• Does friction push the block left? No! Friction only resists the motion.

There are two kinds of friction:

1. Static Friction:
   • Not moving
   • Keeps the object in place (e.g., desk on the floor)
2. Kinetic Friction:
   • While moving
3. Generally, static friction is greater than kinetic friction, making it hard to get the motion started.
4. Friction force does work which often results in the production of HEAT.

Circular Motion

Requires an inward force

• Direction of velocity changes
• \(F=\frac{mv^2}{R}\)
  • Bigger radius, smaller force
  • Bigger velocity, larger force

Water in a bucket

• If the gravitational force is greater than that required for circular motion, the water will fall.
• If the gravitational force is smaller than that required for circular motion, the bucket will exert more force on the water to keep it moving in a circle.

Dishes

• \(F_{gravity} = N\) (block not falling)
• \(F_{friction} =\) kinetic friction (small)
• Get very little acceleration
• \(F_{friction} = ma\)

Note

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ChatGPT said:

Scenario 1: Tablecloth Moves, Cup Moves Slowly

• Tablecloth Movement: The tablecloth is moved to the right (not shown explicitly but implied).
• Friction Force (FfrictionFfriction): Acts to the right. This force arises due to the interaction between the bottom of the cup and the moving tablecloth. The friction tries to drag the cup along with the tablecloth.

• Gravity (FgravityFgravity): Pulls the cup downward, which is balanced by the normal force (N) exerted by the tablecloth upward on the cup.

  N

• Cup Movement: The cup moves to the right but at a slower rate than the tablecloth, depending on the friction coefficient and the speed of the tablecloth's movement.

Scenario 2: Tablecloth Stops, Cup Comes to a Stop

• Tablecloth Stops: Suddenly ceases to move.
• Friction Force (FfrictionFfriction): Acts to the left, opposite to the previous direction of motion. This happens because, upon stopping the tablecloth, the cup still has momentum and tries to continue moving right. Friction then acts in the opposite direction to slow and eventually stop the cup.
• Gravity and Normal Force: These forces remain the same as in the first scenario, where gravity pulls down and the normal force balances it out.

Key Points

• Friction and Motion: In both scenarios, friction acts against the relative motion between the cup and the tablecloth. Initially, it tries to move the cup along with the tablecloth, and later, it works to stop the cup when the tablecloth is no longer moving.
• Effectiveness of Friction: The effectiveness of friction in moving or stopping the cup depends on factors like the speed of the tablecloth’s movement, the mass of the cup, and the coefficient of friction between the cup and the tablecloth.

Note

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ChatGPT said:

When a person sitting on a spinning chair extends their arms, their rotational speed decreases due to the conservation of angular momentum. Here’s how it works:

1. Angular Momentum Conservation: In a closed system with no external torques, angular momentum is conserved. This means that the total angular momentum before and after a change in body position (like extending arms) remains constant.

2. Moment of Inertia: Angular momentum \(L\) is given by the product of the moment of inertia \(I\) and angular velocity \(\omega\):

   \[ L = I \omega \]

   The moment of inertia depends on how mass is distributed relative to the axis of rotation. When the person pulls their arms close to their body, their mass is concentrated closer to the rotation axis, reducing their moment of inertia. Extending the arms increases the moment of inertia because mass is distributed farther from the axis.

3. Inverse Relationship with Angular Velocity: To conserve angular momentum, when the moment of inertia increases (by extending arms), the angular velocity must decrease to keep \(L\) constant:

   \[ I_{\text{new}} \omega_{\text{new}} = I_{\text{old}} \omega_{\text{old}} \]

   So, if \(I\) increases, \(\omega\) (the spinning speed) decreases.

Why It Slows Down

By extending the arms, the person’s moment of inertia increases, and to conserve angular momentum, their spinning speed (angular velocity) decreases. This effect can be seen with figure skaters or divers who adjust their rotation speed by changing their body position.