Circular Motion and the Law of Gravity
In this Chapter, the quantities needed to describe circular motion will be defined.
These include angular velocity, angular acceleration, tangential velocity and
acceleration and, centripetal acceleration.
Angular Velocity and Acceleration
Definition: Angular Velocity
of a rigid body is the rate of change of its angular position. Thus,
at t = t1 , and =
body over the time interval
at t = t2 , then the average angular velocit
Relationship Between Linear and Angular
Figure 7.2: Circular Motion
Consider an object that moves from point P to P' along a circular trajectory of
radius r , as shown in Figure 7.2.
Definition: Tangential Speed
The average tangentia
Consider an object moving in a circle of radius r with constant angular velocity. The
tangential speed is constant, but the direction of the tangential velocity vector changes
as the object rotates.
Newton's Law of Gravitation
Idea: Newton's Universal Law of Gravitation states that any two objects exert a
gravitational force of attraction on each other. The direction of the force is along the
line joing the objects (See Fig.(7.3). The magn
A bicycle wheel of radius r = 1.5 m starts from rest and rolls 100 m without slipping
in 30 s. Calculate a) the number of revolutions the wheel makes, b) the number of
radians through which it turns, c)The average angular velocity.
= 1.37 rads/s.
The tangential velocity and acceleration are:
vt = r
= (1.5 m )(1.37 rads/s ) = 2.06 m/s
at = r
= (1.5 m )(0.15 rads/s 2) = .225 m/s 2.
A yo-yo of mass 100 g is spun in a vertical circle at the end of a 0.8 m string at a
At the bottom of the trajectory, the centripetal force is provided by the
difference between the tension and the weight (see Figure 7.5).
mac = T - mg
T = mac + mg = mr
+ mg = 2.26 N.
The same yo-yo as in the previous problem is spun in a horizont
This implies that
= 1.13 N.
mac = Tsin
= 3.76 rads/s.
The period is therefore:
= 1.67 s.
90 o , T
in order to balance the gravitational force, so that the
0: i.e. you must
The above expression is called Kepler's Law. Note: The period is independent of the
mass of the satelite.
The escape velocity of any object is the speed it must achieve to escape the
gravitational pull of the Earth. Calculate the escape velocity for an ob