# Statics - 3D equilibrium problem

• cyberdeathreaper
In summary, the question is asking for the vertical components of the reactions at points A, B, and C when the camera is mounted on a tripod at a certain angle, and for the maximum angle at which the tripod can support the camera without flipping over. For part (a), the answer can be easily obtained by finding the sum of all moments at point D. However, for part (b), the critical condition is the distribution of mass relative to the line connecting points A and C. When the line of action of the camera is too far from this line, the tripod will not be able to maintain equilibrium and the maximum angle at which the tripod can support the camera without flipping over is 54.1 degrees.
cyberdeathreaper
Here's the question:

"A camera of mass 240g is mounted on a small tripod of mass 200g. Assuming that the mass of the camera is uniformly distributed and that the line of action of the weight of the tripod passes through D, determine (a) the vertical components of the reactions at A, B, and C when $\theta$ = 0. (b) the maximum value of $\theta$ if the tripod is not to flip over."

(Refer to the attachment to better interpret what is being asked.)

I've gotten part (a) easily (summation of the forces and moment about D easily gives the answer). However, for part (b), I'm loss. As far as I can tell, the z component of the momentum about D is zero only for mutiples of 180 degrees, and yet the answer indicates that the maximum occurs at $\theta = 54.1^o$. Any ideas?

#### Attachments

• phys.jpg
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Note:
Here's the equations and coordinates I'm using (notice that I've moved the coordinate system down to the bottom, so D is at (0,0,0):

A: (-0.045 m, 0, 0)
B: (0.035 m, 0, 0.038 m)
C: (0.035 m, 0, -0.038 m)
D: (0, 0, 0)
F: ( $\alpha , 0, \beta$ )

where F is where the line of action of the weight of the camera intersects the xz plane, and where:

$$\alpha = -0.036 cos( \theta )$$

$$\beta = -0.036 sin( \theta )$$

If you find the sum of all the moments at D, it should equal zero if the object is static.

cyberdeathreaper said:
Note:
Here's the equations and coordinates I'm using (notice that I've moved the coordinate system down to the bottom, so D is at (0,0,0):

A: (-0.045 m, 0, 0)
B: (0.035 m, 0, 0.038 m)
C: (0.035 m, 0, -0.038 m)
D: (0, 0, 0)
F: ( $\alpha , 0, \beta$ )

where F is where the line of action of the weight of the camera intersects the xz plane, and where:

$$\alpha = -0.036 cos( \theta )$$

$$\beta = -0.036 sin( \theta )$$

If you find the sum of all the moments at D, it should equal zero if the object is static.

The critical condition is how the mass is distributed relative to the line AC. When the line of action of the camera gets too far over that line, the moment of the tripod will be insufficient to maintain equilibrium.

## What is the definition of a 3D equilibrium problem?

A 3D equilibrium problem is a type of statics problem that involves determining the forces and moments acting on a three-dimensional object in order for it to remain in a state of balance. This problem is often encountered in engineering and physics, and requires the use of mathematical equations and principles to solve.

## What are the key principles used to solve a 3D equilibrium problem?

The key principles used to solve a 3D equilibrium problem include the laws of equilibrium, which state that the sum of all forces acting on an object must equal zero and the sum of all moments acting on an object must also equal zero. Additionally, the use of free body diagrams and vector analysis are important tools in solving these types of problems.

## What are some common real-world applications of 3D equilibrium problems?

3D equilibrium problems have many real-world applications, including in the design of structures such as buildings, bridges, and airplanes. They are also useful in understanding the stability of objects on inclined planes, calculating loads on joints and supports, and analyzing the forces acting on mechanical systems.

## What are some tips for solving a 3D equilibrium problem?

Some tips for solving a 3D equilibrium problem include drawing accurate and clear free body diagrams, labeling all forces and distances involved, and breaking down complex systems into smaller, more manageable parts. It is also important to pay attention to units and use proper mathematical techniques to solve the equations.

## What are some common mistakes to avoid when solving a 3D equilibrium problem?

Some common mistakes to avoid when solving a 3D equilibrium problem include forgetting to account for all forces and moments acting on the object, using incorrect units, and making calculation errors. It is also important to check the final solution to ensure that it makes sense in the context of the problem and that all units are consistent.

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