A small block of mass m rests on the sloping side of a triangular block of mass M which itself rests on a horizontal table.
Assuming all surfaces are frictionless, determine the magnitude of the force ->F that must be applied to M so that m remains in a fixed position relative to M (that is, m doesn't move on the incline). [Hint: Take x and y axes horizontal and vertical.]
How would do this problem? Why take x and y axes horizontal and vertical?
(M+m)*g*tan(theta)
dont know how i got it but its right
Well, isn't it obvious? Taking the axes horizontal and vertical is much more convenient because if we took them unicorn and rainbow-shaped, the calculations would be quite challenging, don't you think? Plus, horizontal and vertical axes are widely known for their trustworthiness. You can always count on them to keep things in line. So, let's dive into this problem, shall we?
To determine the magnitude of force F, we need to consider the forces acting on both masses, M and m.
Let's start with mass M. Since it's on a horizontal table, the only force acting on it is its weight, which we can represent with Mg, pointing downwards. Easy enough, right?
Now, let's move on to mass m. Since we want it to stay in a fixed position relative to M, it means there should be no net force acting on it along the incline. So, the force of gravity acting on m, mg, needs to be balanced by a component of the force F.
To find that component, we can resolve F into two perpendicular directions: one parallel to the incline and another perpendicular to it. We're only interested in the parallel component, so let's focus on that.
Using our trusty axes, we can write the equation for the net force in the x-direction as follows:
F_parallel - mg*sin(theta) = 0
This equation basically says that the force F component parallel to the incline minus the component of mg in the same direction should add up to zero. This ensures that m remains stationary on the incline.
Simplifying the equation, we get:
F_parallel = mg*sin(theta)
And there you have it. The magnitude of the force F that needs to be applied to M in order to keep m in a fixed position relative to M is equal to mg*sin(theta). Just substitute the given values of m, g, and theta into the equation, and voila!
To solve the problem, we can use the concepts of equilibrium in both the x and y directions. Taking x and y axes horizontal and vertical helps in separating the forces acting on the system into their respective components along the axes. This allows us to analyze the forces in each direction separately, making the problem easier to solve.
To determine the magnitude of the force F that must be applied to M so that m remains in a fixed position relative to M, follow these steps:
Step 1: Identify the forces acting on the system:
- Weight of the small block m (mg), acting vertically downward.
- Normal force from the triangular block M, acting perpendicular to the inclined surface.
- Force F applied horizontally to the triangular block M.
Step 2: Resolve the forces into their x and y components:
- For the weight of m, the vertical component is mg sin(theta) and the horizontal component is mg cos(theta), where theta is the angle of inclination of the sloping side.
- The normal force from M only has a y component, which is equal and opposite to mg sin(theta).
- The applied force F only has an x component.
Step 3: Write the equations of equilibrium:
- In the x direction, there is no acceleration, so the sum of the forces in the x direction is zero:
F = mg sin(theta)
- In the y direction, there is no acceleration, so the sum of the forces in the y direction is zero:
N - mg cos(theta) = 0
Step 4: Solve the equations simultaneously:
From the equation in the y direction, we can solve for N:
N = mg cos(theta)
Substituting this value into the equation in the x direction, we have:
F = mg sin(theta)
Therefore, the magnitude of the force F that must be applied to M is F = mg sin(theta).
Note: It is important to use trigonometry to obtain the correct components of the forces and to recognize that the normal force is equal and opposite to the vertical component of the weight.
To solve this problem, we can break it down into two parts: analyzing the forces acting on the triangular block (M) and analyzing the forces acting on the small block (m). By treating them separately, we can determine the overall force (F) required to keep m in a fixed position relative to M.
Taking the x and y axes horizontal and vertical is a common convention in physics problems. This allows us to resolve forces into their respective horizontal and vertical components, simplifying the analysis of the forces involved. It also allows us to use trigonometry to find the angles and relationships between different forces.
Here's how to approach the problem step by step:
1. Analyze the forces acting on M:
- Draw a free-body diagram for M, showing all the forces acting on it. These forces include the weight of M (Mg) acting vertically downwards and the force F applied to M.
- Resolve the force F into its x and y components. Let's call the x-component Fx and the y-component Fy.
2. Analyze the forces acting on m:
- Draw a free-body diagram for m, showing all the forces acting on it. These forces include the weight of m (mg) acting vertically downwards and the normal force N exerted by the inclined surface of M.
- Since the surfaces are frictionless, there is no friction force.
3. Apply Newton's second law:
- In the x-direction, the force Fx acting on M must counteract the x-component of the force Mg acting on M. The x-component of Mg is given by Mg*sinθ, where θ is the angle of the inclined surface.
- In the y-direction, the force Fy acting on M must counteract the y-component of Mg and the normal force N. The y-component of Mg is given by Mg*cosθ.
- The normal force N is equal to the weight of m (N = mg).
- Equate the forces in the x and y directions to zero and solve for Fx and Fy.
4. Determine the magnitude of F:
- The magnitude of the force F required to keep m in a fixed position is given by F = √(Fx^2 + Fy^2).
By following these steps and applying Newton's second law, you can determine the magnitude of the force (F) that must be applied to M to keep m in a fixed position relative to M.