Solving for Velocity in a Bead's Circular Motion

In summary, the problem involves a bead of mass m moving on a horizontal circular ring of wire with an initial speed v_0 and a coefficient of friction mu_k. The radius of the ring is r and gravity can be neglected. The motion at later times is being investigated, with a focus on finding the equation v(t) by combining the equations for radial and tangential accelerations. The total acceleration can be found by combining these equations and using the relationship between acceleration and velocity to solve for v.
  • #1
Crater
5
0
Given this set-up.


Consider a bead of mass m that is free to move around a horizontal, circular ring of wire (the wire passes through a hole in the bead). You may neglect gravity in this problem (assume the experiment is being done in space, far away from anything else). The radius of the ring of wire is r. The bead is given an initial speed v_0 and it slides with a coefficient of friction mu_k. In the subsequent steps we will investigate the motion at later times. You should begin by drawing a free-body diagram at some instant of time. Note that there will be a radial acceleration, a_R, and a tangential acceleration, a_T, in this problem.


I'm suppose to find an equation v(t) by combining the equations of radial and tangential accelerations.

With a_R = v^2/r and a_T = dv/dt. I've found a_T to also be equal to N*mu_k with N=m*a_R= mu_k*(v^2/r).

Whenever I try to combine any of these equations and solve for v, I can never get it in terms of other values and t. So I'm at a loss. Any ideas?
 
Physics news on Phys.org
  • #2
well, you know aR and aT

so find the total acceleration and knowing that acceleration is the rate of change of velocity, you can use the relationship

a = dv/dt

to solve for v

by rearranging

a*dt = dv

and

integrating
 
  • #3


I would suggest approaching this problem by first writing out all of the known equations and variables, and then manipulating them to solve for the desired variable (in this case, v). It may also be helpful to consider the physical meaning of each equation and how they relate to the motion of the bead.

To begin, we can write out the equations of motion for the bead:

1. Newton's Second Law: F = ma
2. Centripetal Force: F = mv^2/r
3. Frictional Force: F = mu_kN

Next, we can consider the forces acting on the bead at a specific instant in time, as shown in the free-body diagram. The only forces acting on the bead are the normal force (N) and the frictional force (F), since we are neglecting gravity. We can set up an equation for the net force in the radial direction:

F_net,r = mv^2/r - mu_kN = ma_R

We can also set up an equation for the net force in the tangential direction:

F_net,t = ma_T = m(dv/dt)

Using the equation for the frictional force, we can substitute in for N:

F_net,r = mv^2/r - mu_k(ma_R) = ma_R

Now we have two equations with two unknowns (v and a_R). We can solve for a_R in terms of v:

a_R = (v^2/r)(1 - mu_k)

Substituting this into our equation for the net force in the tangential direction, we get:

F_net,t = m(dv/dt) = m[(v^2/r)(1 - mu_k)]

We can now use calculus to solve this differential equation and obtain an equation for v(t):

v(t) = v_0e^(-mu_kt/r)

So the velocity of the bead at any time t is given by this equation, which combines the effects of radial and tangential accelerations. It is important to note that this equation assumes the bead is sliding without slipping on the ring, and that the coefficient of friction remains constant throughout the motion. If these assumptions do not hold, the equation may need to be modified.
 

1. What is circular motion?

Circular motion is the movement of an object along a circular path at a constant speed.

2. How is velocity defined in circular motion?

Velocity in circular motion is defined as the rate of change of an object's position with respect to time in a circular path. It has both magnitude (speed) and direction.

3. How do you calculate velocity in circular motion?

To calculate velocity in circular motion, you need to divide the circumference of the circular path by the time it takes for the object to complete one full revolution. This will give you the object's average velocity. Alternatively, you can also use the formula v = 2πr / t, where v is the velocity, r is the radius of the circular path, and t is the time taken.

4. What factors affect the velocity of a bead in circular motion?

The velocity of a bead in circular motion is affected by the radius of the circular path, the speed at which the object is moving, and the force acting on the object. Additionally, the presence of friction or other external forces can also affect the velocity of the bead.

5. How can you use the equations of circular motion to solve for velocity in a bead's circular motion?

The equations of circular motion, such as v = 2πr / t and a = v^2 / r, can be used to solve for velocity in a bead's circular motion. By plugging in known values for the radius, time, and acceleration, you can solve for the velocity of the bead in the circular path.

Similar threads

  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
909
  • Introductory Physics Homework Help
Replies
14
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
182
  • Introductory Physics Homework Help
Replies
14
Views
2K
  • Introductory Physics Homework Help
2
Replies
44
Views
7K
  • Introductory Physics Homework Help
Replies
16
Views
14K
  • Introductory Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
Back
Top