Dynamics of Uniform Circular Motion Rotational Kinematics Chapter 5 Dynamics of Uniform Circular Motion Chapter 8 Rotational Kinematics

Uniform circular motion A special case of 2D motion An object moves around a circle at a constant speed Period – time to make one full revolution An object traveling in a circle, even though it moves with a constant speed, will have an acceleration

Centripetal acceleration Centripetal acceleration is due to the change in the direction of the velocity Centripetal acceleration is directed toward the center of the circle of motion

Centripetal acceleration The magnitude of the centripetal acceleration is given by

Centripetal acceleration During a uniform circular motion: the speed is constant the velocity is changing due to centripetal (“center seeking”) acceleration centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward

Centripetal force For an object in a uniform circular motion, the centripetal acceleration is According to the Newton’s Second Law, a force must cause this acceleration – centripetal force A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed

Centripetal force Centripetal forces may have different origins Gravitation can be a centripetal force Tension can be a centripetal force Etc.

Centripetal force Centripetal forces may have different origins Gravitation can be a centripetal force Tension can be a centripetal force Etc.

Free-body diagram

Chapter 5 Problem 55 A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If r = 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?

Newton’s law of gravitation Any two (or more) massive bodies attract each other Gravitational force (Newton's law of gravitation) Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant

Satellites Accounting for the shape of Earth, projectile motion has to be modified:

Satellites For a circular orbit and the Newton’s Second law Thus, a speed of a satellite

Chapter 5 Problem 33 A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.70 × 104 m/s, and the radius of the orbit is 5.25 × 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.60 × 106 m. What is the orbital speed of the second satellite?

The radian The radian is a unit of angular measure The angle in radians can be defined as the ratio of the arc length s along a circle divided by the radius r

Rotation of a rigid body We consider rotational motion of a rigid body about a fixed axis Rigid body rotates with all its parts locked together and without any change in its shape Fixed axis: it does not move during the rotation This axis is called axis of rotation Reference line is introduced

Angular position Reference line is fixed in the body, is perpendicular to the rotation axis, intersects the rotation axis, and rotates with the body Angular position – the angle (in radians or degrees) of the reference line relative to a fixed direction (zero angular position)

Angular displacement Angular displacement – the change in angular position. Angular displacement is considered positive in the CCW direction and holds for the rigid body as a whole and every part within that body

Angular velocity Average angular velocity Instantaneous angular velocity – the rate of change in angular position

Angular acceleration Average angular acceleration Instantaneous angular acceleration – the rate of change in angular velocity

Rotation with constant angular acceleration Similarly to the case of 1D motion with a constant acceleration we can derive a set of formulas:

Chapter 8 Problem 26 A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.05 × 104 rad/s to an angular speed of 3.14 × 104 rad/s. In the process, the bit turns through 1.88 × 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 7.85 × 104 rad/s, starting from rest?

Relating the linear and angular variables: position For a point on a reference line at a distance r from the rotation axis: θ is measured in radians

Relating the linear and angular variables: speed ω is measured in rad/s Period

Chapter 8 Problem 41 A baseball pitcher throws a baseball horizontally at a linear speed of 42.5 m/s (about 95 mi/h). Before being caught, the baseball travels a horizontal distance of 16.5 m and rotates through an angle of 49.0 rad. The baseball has a radius of 3.67 cm and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the “equator” of the baseball?

Relating the linear and angular variables: acceleration α is measured in rad/s2 Centripetal acceleration

Total acceleration Tangential acceleration is due to changing speed Centripetal acceleration is due to changing direction Total acceleration:

Third Kepler’s law For a circular orbit and the Newton’s Second law From the definition of a period Johannes Kepler (1571-1630)

Chapter 5 Problem 53 Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of satellite A is three times that of satellite B. Find the ratio (TA/TB) of the periods of the satellites.

Questions?