A bead with a hole through it slides on a wire track.

Question

A bead with a hole through it slides on a wire track. The wire is threaded through the hole in the bead, and the bead slides without friction around a loop-the-loop (see figure below). The bead is released from rest at a height

h = 3.30R.

A bead is on a loop of wire at height h. The wire curves down and to the right, then loops into a complete circle of radius R before continuing on to the right. The highest point of the circle is labeled A. Position A is lower than the initial position of the bead.

(a)

What is its speed at point A? (Use the following as necessary: the acceleration due to gravity g, and R.)

V =

(b)

How large is the normal force on the bead at point A if its mass is 4.50 grams?

magnitude:

___ N

direction:

---Select--- upward or downward

(c)

What If? What is the minimum height h from which the bead can be released if it is to make it around the loop? (Use any variable or symbol stated above as necessary.)

h =

Answer 1

Since there is no friction, the mechanical energy (Kinetic + Potential) will be conserved.

You will also have to use Newton's second law to find the normal force on the bead at the highest point of the circle.

(a) Conservation of Kinetic + Potential energy:

K_i + U_i = K_f + U_f

0 + mgh = mV²/2 + mg(2R)

V = root(2g(h - 2R))

V = root(2g(3.30R - 2R))= root(2.60gR)

(b) Newton's second law at the highest point of the circle:

sum of the vertical forces on the bead = ma_centripetal

F_n - mg = - mV²/R (negative signs because mg and the centripetal acceleration point downwards)

F_n = m(g - V²/R)

F_n = m(g - 2.60g) = - 1.60mg (downwards since it comes out negative)

(c) The minimal height is when the bead reaches the highest point on the circle with a speed zero:

From part (a):

V = root(2g(h - 2R)) = 0

h = 2R

So the bead has to start at the same height as the highest point on the circle.

Note: in that setup, the wire can provide any normal force F_n , towards or away from the center of the circle. Finding the minimal height h by setting F_n = 0 is invalid. It would be valid if the bead was sliding on the track, not threaded through it, and then the normal force would always have to be towards the center of the circle, so it has to be negative (pointing downwards) at the highest point, or zero in the case of minimal height h.