Motion in a non-inertial frame

Linear acceleration

Problem:

At t = 0 a 100 g ball is thrown upward with initial speed v = 2 m/s in an open-platform elevator which at that time is moving downward with v = 3 m/s and accelerating downward with an acceleration of magnitude a = 3 m/s2.  A drag force with magnitude Fdrag = 0.8 N acts on the ball. 
(a)  What is the net force Fnet acting on the ball in the frame of the elevator just after it has been thrown?
(b)  What is the net force Fnet acting on the ball in the frame of a person standing on the ground just after it has been thrown?

Solution:

Problem:

A hauling truck is traveling on a level road. The driver suddenly applies the brakes, causing the truck to decelerate by an amount g/2.  This causes a box in the rear of the truck to slide forward. If the coefficient of sliding friction between the box and the truck bed is ⅓, find the acceleration of the box relative to
(a) the truck and
(b) the road.

Solution:

Problem:

(a)  An elevator in which a woman is standing moves upward at 4 m/s.  If the woman drops a coin from a height 1.4 m above the elevator floor, how long does it take the coin to strike the floor?  What is the speed of the coin relative to the floor just before impact? 
(b)  Now assume that the elevator is moving downward with zero initial velocity and acceleration of 1 m/s2 at t = 0, the women releases the coin at t = 1 s.  How long does it take the coin to strike the floor?  What is the speed of the coin relative to the floor just before impact?

Solution:


Rotating frames

Problem:

Write F = ma in a rotating coordinate system and identify Coriolis and centrifugal terms.

Solution:

Problem:

A disc of radius R is spinning in the horizontal plane with a constant angular speed Ω.  A ladybug walks along the radius of the spinning disc, traveling from the center of the disc toward the edge. The ladybug maintains a constant speed v relative to the disc.
What is the acceleration of the ladybug at the instant it reaches the edge of the disc?

Solution:

Problem:

An object on a planar platform moves in an elliptical trajectory described by
x(t) = x1 + x2 cos(αt),   y(t) = y1 + y2 sin(αt),
where x and y are measured with respect to a coordinate system fixed on the platform.  The platform is rotating with respect to an inertial coordinate system XYZ.  The two coordinate systems XYZ and xyz are the same at time t = 0.  The axis of rotation is the Z-axis, but the angular speed of rotation is fluctuating with time and is given by
ω(t) = ω1 + ω2 sin(βt).
Find an expression for the velocity components Vx and Vy of the object in the inertial frame at time.

Solution:

Problem:

A brave physics student (an undergraduate, of course) climbs aboard a high powered merry-go-round and goes to the center, at r = 0.  At time t = 0, the platform starts from rest (Ω = 0) and begins to spin about its vertical axis with constant angular acceleration α.  Also at time t = 0, the student begins to crawl radially outward at constant speed v, relative to the platform.
Assuming the student does not slip, find the student's acceleration in the inertial frame of an outside observer.

Solution:

Problem:

On the surface of the earth an object is given an initial speed v on a friction less surface at latitude λ.  Show that the object will move in a circle and find the radius of the circle for velocities small enough that the radius is much smaller than the earth radius.

Solution:

Problem:

A particle moves in a horizontal plane on the surface of the Earth.  Show that the magnitude of the horizontal component of the Coriolis force is independent of the direction of the motion of the particle.

Solution: