Mechanics Problem (falling object)

In summary: We can solve for x by integrating from 0 to d:x=\int_0^d \! x \, \mathrm{d} m=\frac{g\cdot I}{m}This gives us the approximate speed of the highest point of the ladder as it hits the ground.
  • #1
infke
1
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Hello, let me start with saying this is not homework but rather a question I have asked myself and cannot seem to solve.

Homework Statement


Basically this:

If a ladder is standing against a wall (both in a 90° corner with the ground) and falls over, what is the approximate speed of the highest point of the ladder as it hits the ground? The only force you should take note of is the gravitational acceleration. Because of the state of the ladder it shouldn't move according to the laws of physics but let's assume you give it a neglectable push to start the movement. You can also assume that the part touching the ground does not move.

The problem with this is that you don't have a linear acceleration. It varies anywhere from 0 to g (9.81m/s²) and I have no idea how to solve this correctly.

Homework Equations


The distance (x) the highest point has traveled over is
x = r * ∏/2
with r being the length of the ladder

Also the acceleration along the y curve on any given point A on the x curve smaller than r
a = g * cos ( A / r )

The Attempt at a Solution


Because the acceleration is not linear I did not find a correct way to calculate my question.
 
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  • #2
This problem requires some calculus (integration) if you want to find the final velocity of the ladder. I'll tell you how to find how fast the ladder is accelerating for a given angle, and if you want, we can go on to the integration from there.

You can find the torque at the ladder's pivot point by using the equation:
[itex]τ=r_{\perp}F[/itex] or [itex]τ=rFsinϕ[/itex]
Where ϕ is the angle between the gravity vector and the axis along the ladder's length.
You can assume that the force is being applied at the ladder's center of mass (so the length of r is equal to half the ladder's height).

Using Newton's second law for rotation, we can use this torque to determine the angular acceleration:
[itex]\large α=\frac{τ}{I}[/itex]
I is the ladder's moment of inertia.
If we assume the ladder to be a thin rod, the moment of inertia for rotation about the end is:
[itex]I=\frac{1}{3}ML^{2}[/itex]

If we substitute τ and I into the rotation equation, we get:
[itex]\large α=\frac{3rFsinϕ}{ML^{2}}[/itex]
We can make two more substitutions: F=Mg and r=L/2
[itex]\large α=\frac{3}{2}\frac{LMgsinϕ}{ML^{2}}[/itex]
Now we can cancel out an L and M from both sides of the fraction.
[itex]\large α=\frac{3}{2}\frac{gsinϕ}{L}[/itex]
Since ϕ is a bit of an odd angle, let's substitute ϕ=θ+90°, making θ the angle between the ground and the ladder.
[itex]\large α=\frac{3}{2}\frac{gsin(θ+90)}{L}[/itex]
and we can substitute sin(θ+90)=cosθ to simplify it a bit more:
[itex]\large α=\frac{3}{2}\frac{gcos(θ)}{L}[/itex]
This equation gives the angular acceleration for a ladder of length L at an angle θ from the ground. Since the angle in this equation is a function of time, this is where you'd need calculus to move on with the problem.
 
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  • #3
Using conservation of energy, PE=KE
PE=mgh
[itex]PE=g\int_0^d \! x \, \mathrm{d} m[/itex]
KE=0.5Iω2
 

FAQ: Mechanics Problem (falling object)

What is the equation for the velocity of a falling object?

The equation for the velocity of a falling object is v = gt, where v is the velocity in meters per second, g is the acceleration due to gravity (9.8 m/s^2), and t is the time in seconds.

How does air resistance affect the speed of a falling object?

Air resistance, also known as drag, acts in the opposite direction of the object's motion and increases as the speed of the object increases. This means that as a falling object accelerates, the force of air resistance also increases, eventually balancing out the force of gravity and causing the object to reach a terminal velocity (constant speed). Therefore, air resistance can limit the speed of a falling object.

What is the difference between weight and mass?

Weight is a measure of the force of gravity pulling on an object, while mass is a measure of the amount of matter in an object. Mass is constant and does not change based on location, but weight can change depending on the strength of gravity. For example, an object will have the same mass on Earth and on the moon, but its weight will be different due to the difference in gravitational pull.

How does the height from which an object is dropped affect its acceleration?

The height from which an object is dropped does not affect its acceleration due to gravity. All objects, regardless of their mass or height, will accelerate towards the ground at a rate of 9.8 m/s^2 (ignoring air resistance). However, the object's initial velocity will be greater if it is dropped from a higher height.

Can an object have a negative velocity while falling?

Yes, an object can have a negative velocity while falling if it is thrown upwards or if it is falling with a downward acceleration (e.g. a parachute). This means that the object is moving in the opposite direction of the acceleration due to gravity, but it is still considered to be falling as it is moving towards the ground.

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