# What happens when a bug lands on a spinning sphere?

• Epiphone
In summary, two identical spheres of mass M and negligible radius are attached to a rod of negligible mass and length 2l, free to rotate about a frictionless horizontal axis. A bug of mass 3M lands on one of the spheres, causing the system to rotate. The torque immediately after the bug lands is 3Mgl. The angular acceleration of the system is 9g/16Ml and the angular speed of the bug is sqrt(9(pi)g/16Ml). The angular momentum of the system is 4Ml^2(sqrt(pi)g/Ml). To keep the bug from being thrown off the sphere, a force of 27g/16M must be exerted on it. The
Epiphone

## Homework Statement

Two identical spheres, each of mass M and negligible radius, are fastened to opposite ends of a rod of negligible mass and length 2l. This system is initially at rest with the rod horizontal and is free to rotate about a frictionless, horizontal axis through the center of the rod and perpendicular to the plane of the page. A bug of mass 3M, lands gently on the sphere on the left. Assume that the size od the bug is small compared to the length of the rod. Express your answers to all parts of the question in terms of M, l, and physical constants.

a.) Determine the torque about the axis immediately after the bug lands on the sphere.

b.) Determine the angular acceleration of the rod-spheres-bug system immediately after the bugs lands.

At the instant that the rod is vertical (pointing up and down meaning it has rotated 90 degrees counterclockwise) determine each of the following:

c.) The angular speed of the bug.

d.) The angular momentum of the system

e.) The MAGNITUDE and DIRECTION of the force that must be exerted on the bug by the wphere to keep the bug from being thrown off the sphere.

T = r x F
T = Ialpha
L = r x p
L = Iw

## The Attempt at a Solution

I realize this question has been asked on the forum before, but i have not found the other posts very helpful because they needed help with different parts

here are my solutions so far:
a) T = 3Mgl

b) alpha = 9g/16Ml
(solved by setting T = r x F = Ialpha)

c) w = sqrt(9(pi)g/16Ml)
(solved using w^2 = w^2 + 2alpha...)

d) L = 4Ml^2(sqrt(pi)g/Ml)
(solved by L = Iw)

e) I am stuck here! i don't know what concept to use!

Hint: What's the bug's acceleration?

hm, since alpha = a/l
alpha = 9g/16Ml
so a = 9g/16M?

and f = ma, so f = (3M*9g)/16M = 27g/16M?

i feel like i am making up rules here :/

Epiphone said:
hm, since alpha = a/l
alpha = 9g/16Ml
so a = 9g/16M?
Two problems here:
(1) alpha will just give you angular acceleration (and tangential); what about radial acceleration?
(2) that's alpha at the initial position, not when the system is vertical.

Another hint: What kind of motion is the bug exeriencing at the lowest point?

i think radial is V^2/r

im a little confused though. why would it be at the "lowest point" if the rotational axis is perpendicular to the page? isn't the height always the same?

its instantaneous velocity is tangent to the circular path and perpendicular to the radius, but I am not sure if that is the type of motion you were referring to in your hint

and does this problem make my answer to part c wrong? because i used the initial alpha in my calculation of w^2 = w^2 + 2alpha(delta(theta))

Epiphone said:
i think radial is V^2/r
Exactly. What's the bug's speed?

im a little confused though. why would it be at the "lowest point" if the rotational axis is perpendicular to the page? isn't the height always the same?
The axis remains at the same height, but not the masses. Certainly not the bug.

its instantaneous velocity is tangent to the circular path and perpendicular to the radius, but I am not sure if that is the type of motion you were referring to in your hint
Good. And the motion (and acceleration) of the bug is particularly simple to analyze when the rod is vertical.

Epiphone said:
and does this problem make my answer to part c wrong? because i used the initial alpha in my calculation of w^2 = w^2 + 2alpha(delta(theta))
That's certainly not true. (Sorry, I didn't even check your other answers.) Hint for this part: Use conservation of energy.

Doc Al said:
The axis remains at the same height, but not the masses. Certainly not the bug.

right right, for some reason i thought i was looking at the problem from above. that clears up some things!

and for the conservation of energy, initially i think there is only grav potential, and when it is at its lowest point, there is rotational kinetic
so that would mean mgh = .5Iw^2 + mgh
and then i got w = sqrt(9g/8l)

Last edited:
Epiphone said:
and for the conservation of energy, initially i think there is only grav potential, and when it is at its lowest point, there is rotational kinetic
Exactly.
so that would mean mgh = .5Iw^2 + mgh
and then i got w = sqrt(9g/8l)
I don't understand how you got this answer. What's the change in gravitational PE? What's the rotational inertia of the system?

(You'd better redo part b while you're at it.)

## 1. What is angular momentum?

Angular momentum is a measure of an object's rotational motion, specifically how fast it is rotating and in what direction.

## 2. How is angular momentum calculated?

Angular momentum is calculated by multiplying the object's moment of inertia (a measure of how difficult it is to change its rotational motion) by its angular velocity (how fast it is rotating).

## 3. What are some real-life examples of angular momentum?

Some examples of angular momentum in everyday life include the spinning of a top, the rotation of a bicycle wheel, and the movement of a figure skater during a spin.

## 4. How does angular momentum relate to conservation of momentum?

Angular momentum follows the same principles of conservation as linear momentum, meaning that it cannot be created or destroyed, only transferred between objects or changed in direction or speed.

## 5. How is angular momentum important in physics and engineering?

Angular momentum plays a crucial role in many physical phenomena, including the stability and motion of rotating objects, the behavior of celestial bodies, and the functioning of mechanical systems such as engines and turbines. It is also used in engineering to design and optimize structures and machines that involve rotational motion.

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