Solving Object's Velocity at Bottom of Frictionless Incline

In summary, the problem involves an object on a frictionless incline with a changing angle and a constant rate of change. The goal is to find the object's velocity at the bottom of the incline in terms of its length, initial angle, and gravitational constant. This cannot be solved using the basic equation of motion, so it requires setting up and integrating the equations of motion. The final answer is (√Lg(1-cos(θ)))/(√(sin⁡〖θ_o-θ_o cos⁡〖θ_0 〗 〗 ).
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GinnyG
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Homework Statement


An object rests at the top of a frictionless incline with length L and angle θo. At the moment the object is released the angle begins to decrease at a constant rate w. Thus the angle as a function of time is θ(t)= θo-wt. Value of w is defined to be the rate such that at the instant the object reaches the bottom of the incline, θ(t)=0. Find the objects velocity when it reachs the bottom of the incline in terms of L, θo, and the gravitational constant.


Homework Equations



〖Vf〗^2=[Vo]^2+2a∆x

The Attempt at a Solution


I tried solving this by using the equation 〖Vf〗^2=[Vo]^2+2a∆x, with Vo=0 and ∆x=L.
I solved for a by adding up the forces moving in the x direction giving me: a= gsinθ. I plugged this in and got the answer Vf=(2(gsin(θ))(L))^(1/2) but I was wrong. The answer is (√Lg(1-cos(θ)))/(√(sin⁡〖θ_o-θ_o cos⁡〖θ_0 〗 〗 )
I don't understand where to even begin with this problem, please help me!
 
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  • #2


You have to set up and integrate the equations of motion. The equation you are using only applies to constant acceleration.
 

1. How is an object's velocity at the bottom of a frictionless incline calculated?

The velocity of an object at the bottom of a frictionless incline can be calculated using the formula v = √(2gh), where v is the velocity, g is the acceleration due to gravity, and h is the height of the incline.

2. Why is friction not considered when calculating an object's velocity at the bottom of a frictionless incline?

Friction is not considered because a frictionless incline assumes that there is no resistance or force acting against the object as it moves down the incline. Without friction, the object can continue to gain velocity without any external forces slowing it down.

3. How does the angle of the incline affect an object's velocity at the bottom?

The angle of the incline affects an object's velocity at the bottom by altering the height of the incline, which is a key variable in the velocity formula. As the angle increases, the height of the incline decreases, resulting in a lower velocity at the bottom.

4. What is the relationship between an object's mass and its velocity at the bottom of a frictionless incline?

The object's mass does not directly affect its velocity at the bottom of a frictionless incline. However, a heavier object may have a greater potential energy at the top of the incline, resulting in a higher velocity at the bottom.

5. Can an object have a negative velocity at the bottom of a frictionless incline?

Yes, an object can have a negative velocity at the bottom of a frictionless incline if it is traveling in the opposite direction of the incline. This can occur if the object has enough velocity to travel up the incline, but is slowed down by gravity and eventually reaches a negative velocity at the bottom.

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