Lecture Notes

Prepared by:

ENGR. HAROLD JAN R. TERANO, ECE

Lesson 5

ROTATIONAL KINEMATICS AND DYNAMICS

Uniform Circular Motion – an object moves at a constant speed along a circular path.

Velocity is always tangent to the path in circular motion. Speed is constant, velocity is not.

Centripetal Acceleration, – acceleration that maintains the object along a circular path directed towards the center. Also called as radial acceleration.

In 1673, Christian Huygens, determined the following relationships.

Velocity,

Where, r = radius of curvature/path, t = time/period.

Frequency (f) – number of revolutions of cycle completed per unit time.

So,

Expressing centripetal acceleration in terms of frequency,

In terms of period,

Example(1)

It takes a merry-go-round moves 30 seconds to complete one revolution, what is the velocity of the child on top of a horse found 3 meters away from the center? Given:

r = 3 meters

t = 30 seconds

Req’d:

v = ?

Solution:

Example(2)

A micro compact disc (CD) is 6 cm in diameter. If a drive spins uniformly at 300 revolutions per minute, what is the acceleration in m/s2 of a particle of dirt found along the edge? Given:

d = 6 cm

f = 300 rev/min.

Req’d:

ac = ?

Solution:

Centripetal and Centrifugal Forces

Centripetal Force – is the force (real force) on the body towards the center of rotation when a body is moving around a curved path.

Centrifugal Force – is the force (apparent force) on the body directed away from the center of rotation when a body is moving around a curved path.

Where, m = mass in kg, VT = tangential velocity in m/s,

r = radius of curvature

Example(1)

An automobile weighs 1500 kg. If this car is driven around a curve, which has a radius of 250 m at the rate of 45 m/s, what is the centripetal force of the road on the automobile? Given:

m = 1500 kg

r = 250 m

VT = 45 m/s

Req’d:

Fc = ?

Solution:

Banking of Highway Curves

In order for the weight of the car to help the frictional force resist the centrifugal force, banking of highway curve is necessary. Once banked, a car can move at higher speed due to bigger resistance of the centrifugal force.

The ideal angle of banking, θ may be calculated using,

For greater velocity (w/ friction)

Where, , = angle of friction

= coefficient of friction

Example(1)

What is the angle of banking of an elevated highway curve if a car is moving at a speed of 35 kph when the radius of the curve is 50m. Given:

v = 35 kph

r = 50 m

Req’d:

θ = ?

Solution:

Example(2)

Determine the velocity of a car running at a 100 elevated highway with a 25 m radius of path. The coefficient of friction between the tire and the road is 0.93. Given:

r = 25 m

θ = 100

µ = 0.93

Req’d:

v = ?

Solution:

Rotation – is a motion along a circular path.

Where,

S = linear distance in meters

V = linear velocity in m/s

a = linear acceleration in m/s2

θ = angular distance in radians

ω = angular velocity in radians/sec

α = angular acceleration in radians/s2

For variable angular acceleration, the following formulas may be used.

Where,

ω0 = initial angular velocity

ω = final angular velocity

θ = angular distance

α = angular acceleration

t = time

Angular displacement (θ) – is the angle through which a rigid object rotates about a fixed axis. SI unit: radian(rad)

Radian – is the SI unit of displacement defined as the circular arc length (S) traveled by point of a rotating body divided by the distance (r) of the point from the axis.

2π rad = 360 degrees = 1 rev

Average angular velocity (ωave) – is the angular displacement of an object divided by the time elapse. SI unit: rad/sec...