Circular motion with decreasing radius

In summary, to solve this problem, we use the conservation of angular momentum and the equation v_0 r_0 = v_1 r_1 to calculate the time it takes for a ball tied to a vertical cylinder with radius R to strike the cylinder when moving horizontally with initial velocity v0 and initial length of rope L0. Using this equation, we can determine that the final angular velocity is 0 and use the formula time = (final angular velocity - initial angular velocity) / angular acceleration to find the time, where the angular acceleration is the friction constant.
  • #1
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Homework Statement


A ball is tied around a vertical cylinder with radius R by an unstrechable rope, moving around the cylinder horizontally with initial velocity v0 and initial length of rope from the tangent of cylinder L0. How long the ball takes to strike the cylinder? (the friction is constant)


Homework Equations


Conservation of angular momentum, [tex]v_0 r_0 = v_1 r_1[/tex]


The Attempt at a Solution



In my steps, by taking the tangent of the rope on the cylinder as centre, the initial radius of rotation is L0, but the final one is 0 by this way? What is the wrong of my concept?
 
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  • #2
No, that's not correct. The ball will continue to move around the cylinder in a circular motion, so the final angular velocity and final radius of rotation will be the same as the initial angular velocity and initial radius of rotation. To solve this problem, you need to use the conservation of angular momentum. This states that the product of the initial angular velocity (v0/r0) and the initial radius of rotation (r0) is equal to the product of the final angular velocity (v1/r1) and the final radius of rotation (r1). Therefore, we have: v0/r0 * r0 = v1/r1 * r1 Using this equation, we can calculate the time it takes for the ball to strike the cylinder. We know that the initial velocity (v0) and the initial radius of rotation (r0) are given. Therefore, the only unknowns are the final angular velocity (v1) and the final radius of rotation (r1).We can calculate the final angular velocity (v1) by rearranging the equation to give: v1 = (v0/r0) * r1 The final radius of rotation (r1) is simply 0, since the ball will eventually strike the cylinder. Therefore, the final angular velocity (v1) is equal to (v0/r0) * 0, which is 0. Now, we can calculate the time it takes for the ball to strike the cylinder. This can be done using the equation: time = (final angular velocity - initial angular velocity) / angular acceleration where the angular acceleration is equal to the friction constant. Therefore, the time it takes for the ball to strike the cylinder is equal to: time = (0 - v0/r0) / friction constant Hope this helps!
 
  • #3


Your concept is not necessarily wrong, but it may not be the most accurate way to approach this problem. In circular motion, the radius of rotation is constantly changing, but the angular momentum is conserved. This means that the product of the initial velocity and radius (v0r0) must be equal to the product of the final velocity and radius (v1r1).

In this problem, we know the initial velocity (v0) and the initial radius (L0), but we do not know the final radius (r1). We can use this equation to solve for r1 and then use the formula for the circumference of a circle (2πr) to find the distance the ball travels before it strikes the cylinder.

To find the time it takes for the ball to strike the cylinder, we can use the equation for circular motion with constant acceleration (θ = ω0t + 1/2αt^2), where θ is the angle of rotation, ω0 is the initial angular velocity, and α is the angular acceleration. We can solve for t by setting θ equal to 2π (since the ball will complete one full revolution before striking the cylinder) and plugging in the values for ω0 and α, which can be found using the equations ω = v/r and α = a/r, where a is the centripetal acceleration.

Overall, the key to solving this problem is to use the conservation of angular momentum and the equations for circular motion with constant acceleration.
 

1. What is circular motion with decreasing radius?

Circular motion with decreasing radius refers to the movement of an object in a circular path where the distance from the center of the circle decreases over time. This can happen when an external force, such as gravity, acts on the object and pulls it towards the center of the circle.

2. What causes circular motion with decreasing radius?

Circular motion with decreasing radius is caused by an unbalanced force acting on an object in circular motion. This force, such as gravity or friction, pulls the object towards the center of the circle, causing the radius to decrease.

3. How does circular motion with decreasing radius affect an object?

Circular motion with decreasing radius can cause an object to accelerate, meaning its speed increases as it moves towards the center of the circle. This acceleration is caused by the change in direction of the object's velocity, which is constantly pointing towards the center of the circle.

4. What is the relationship between the radius and speed in circular motion with decreasing radius?

In circular motion with decreasing radius, as the radius decreases, the speed of the object increases. This is due to the conservation of angular momentum, where the product of an object's mass, velocity, and radius remains constant.

5. How does circular motion with decreasing radius differ from circular motion with constant radius?

Circular motion with decreasing radius is different from circular motion with constant radius in that the object's speed and acceleration are constantly changing in the former, while they remain constant in the latter. Additionally, circular motion with decreasing radius requires an external force to maintain the motion, while circular motion with constant radius can occur without any external forces acting on the object.

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