A small bead of mass m is constrained to move along a frictionless track as shown. The track consists of a semicircular portion of radius R followed by a straight part. At the end of the straight portion there is a horizontal spring of spring constant k attached to a fixed support. At the top of the circular portion of the track, the bead is pushed with an unknown speed v_0 . The bead comes momentarily to rest after compressing the spring a distance d . The magnitude of the acceleration due to the gravitational force is g . What is the magnitude of the normal force exerted by the track on the bead at the point A, a height R above the base of the track? Express your answer in terms of m , k , R, D , and g but not in terms of V_0 .

To find the magnitude of the normal force exerted by the track on the bead at point A, we need to use the principles of energy conservation.

First, let's consider the initial kinetic energy of the bead when it is pushed with an unknown speed v0 at the top of the circular portion of the track. The kinetic energy is given by:

KE_initial = (1/2) * m * v0^2

Next, let's consider the potential energy of the bead at point A. At this point, the bead is a height R above the base of the track, so the potential energy is given by:

PE_atA = m * g * R

Since the track is frictionless, the only forces acting on the bead at point A are the gravitational force and the normal force. The normal force acts perpendicular to the track surface and balances the gravitational force.

Now, using the conservation of energy principle, we can equate the initial kinetic energy with the potential energy at point A, and solve for the unknown normal force.

KE_initial = PE_atA

(1/2) * m * v0^2 = m * g * R

Simplifying and rearranging the equation, we have:

v0^2 = 2 * g * R

Now, let's move on to the second part of the problem. The bead moves along the straight portion of the track until it comes to rest after compressing the spring a distance d. At this point, all the initial kinetic energy is converted into potential energy of the compressed spring.

The potential energy stored in the compressed spring is given by:

PE_spring = (1/2) * k * d^2

Setting the initial kinetic energy equal to the potential energy stored in the spring, we have:

(1/2) * m * v0^2 = (1/2) * k * d^2

Finally, to find the magnitude of the normal force at point A, we need to consider the balance of forces. At point A, the net force acting on the bead is the sum of the gravitational force and the spring force. Since the bead is momentarily at rest, the net force must be zero.

The gravitational force acting on the bead is m * g, and the spring force is given by Hooke's Law:

F_spring = -k * d

Equating the two forces and considering their magnitudes, we have:

m * g = k * d

Solving for d, we get:

d = (m * g) / k

Substituting the value of d back into the expression for the potential energy stored in the spring, we have:

PE_spring = (1/2) * k * ((m * g) / k)^2

PE_spring = (1/2) * (m^2 * g^2) / k

Now, substituting the value of v0^2 from earlier, we can express the magnitude of the normal force at point A in terms of m, k, R, d, and g:

Magnitude of Normal Force at A = m * g + (k * ((m * g)/k)^2) / (2 * g * R)