Finding energy from graph area and calculating cosθ in two problems

[haha1234]

I have tried to solve this two question.
For question 13, I've plotted a graph.Is it correct?
But I can't find the answer by calculating the area under the line.

For question 11,What is the cosθ

 

[CAF123]

You are plotting F(x) = 2x + 4, which has a non zero F intercept. But your method of finding the work done between those two values of x is correct. Alternatively you could just do the computation W = ∫Fdx, which is equivalent to your method.

In the next question, cosθ represents the cosine of the angle between the force and displacement vectors.

 

[haha1234]

You are plotting F(x) = 2x + 4, which has a non zero F intercept. But your method of finding the work done between those two values of x is correct. Alternatively you could just do the computation W = ∫Fdx, which is equivalent to your method.

For this question, why the change in potential energy is not equal to zero?

 

[CAF123]

For this question, why the change in potential energy is not equal to zero?

Why do you think it should be zero?

 

[haha1234]

Why do you think it should be zero?

Since the object only moves in x direction:shy:

 

[CAF123]

Since the object only moves in x direction:shy:

The key point here is that the force is a function of x. Since V = -W, and W was non zero, we already know that the change in potential energy is similarly non zero.

If you consider an object acted on upon by gravity moving only along the x axis. Gravity points downwards so the angle between the force vector and displacement is 90 degrees and hence W = F.dx = 0 and correspondingly the change in potential energy of the object is zero. The force in this case, though, is not a function of x.

 

[haha1234]

The key point here is that the force is a function of x. Since V = -W, and W was non zero, we already know that the change in potential energy is similarly non zero.

If you consider an object acted on upon by gravity moving only along the x axis. Gravity points downwards so the angle between the force vector and displacement is 90 degrees and hence W = F.dx = 0 and correspondingly the change in potential energy of the object is zero. The force in this case, though, is not a function of x.

Sorry,but what is v?
And how to find the potential energy?

 

[CAF123]

Sorry,but what is v?

V is potential energy here, perhaps you use U.

To elaborate on my example earlier, a block attached to a spring undergoes a restoring force F= -kx when it is displaced some x. We can talk about the potential energy stored in the spring even though the problem is 1 dimensional.

 

[haha1234]

You are plotting F(x) = 2x + 4, which has a non zero F intercept. But your method of finding the work done between those two values of x is correct. Alternatively you could just do the computation W = ∫Fdx, which is equivalent to your method.

In the next question, cosθ represents the cosine of the angle between the force and displacement vectors.

How to find the cosθ?
Is the direction of F show as follow?
And what is the direction of the displacement vector?

 

[nasu]

The work of what force is required by the problem?

 

[CAF123]

How to find the cosθ?

cosθ is the angle between the force vector acting on the particle and and the displacement vector of the particle. The question asks about one particular force and so you have to first identify it's direction. It is not as you have shown it.

And what is the direction of the displacement vector?

Consider first the path OAC. This is made of two segments OA and AC. The particle travels from O to C. Consider each segment separately.