A 4-kg sphere rests on the smooth parabolic surface....

$ A $4-\mathrm{kg}$ sphere rests on the smooth parabolic surface. Determine the normal force it exerts on the surface and the mass $m_{B}$ of block $B$ needed to hold it in the equilibrium position shown.

Key Concepts

Tension and Constraint Forces

In systems where an object is maintained in an equilibrium position by an additional mass or connection, the tension force (or similar constraint force) plays a key role. It provides the necessary counterforce, which must be incorporated into the equilibrium analysis to determine parameters such as the required mass of the attached block.

Curved Surface Geometry

When dealing with curved surfaces, understanding the local geometry is crucial. The tangent and normal to the surface at a point dictate how gravitational and other forces are resolved, influencing the magnitude and direction of reaction forces like the normal force.

Static Equilibrium

This concept refers to a state in which all forces acting on an object balance each other so that the net force is zero. In problems involving objects on surfaces or connections to other masses, identifying static equilibrium conditions is essential for setting up equations that ensure the object remains at rest.

Normal Force

The normal force is the perpendicular contact force exerted by a surface to support an object and prevent it from passing through. In situations where an object rests on a curved surface, the normal force acts along the line normal (perpendicular) to the surface at the point of contact.

Force Decomposition

Force decomposition involves breaking a force into its components along chosen directions, usually aligning one axis with the surface of contact. This approach simplifies the analysis of equilibrium by allowing the independent balancing of forces in perpendicular directions.

Transcript

00:01 In this question, we have this setup and then we have 4kg sphere, press on a smooth parabolic surface, and then everything is in equilibrium.

00:12 So we want to find the mass of the block, that kind of supports the ball, as well as the normal force acting on the, on the ball.

00:27 Okay, so when we do this problem because we are we need to determine the normal force.

00:37 So one thing is we really need to find out like what is the tangent line, what's the angle of the tangent line make with the horizontal? okay, at this point.

00:53 So to do this, so the angle, tangent data is actually the gradient of the line that's tangent to the curve okay and then so the curve has a function y equals to 2 .5 x square so we are going to do divide the x okay at x equals to 0 .4 and when you do a divide the x you get 5x okay and x at x equals to 0 .4.

01:34 Okay, so when you do this, you get two, okay, which means that data is 63 .43 degrees.

01:44 Okay, data is here.

01:46 Okay.

01:48 And then based on the given information, if you draw the free body diagram for the ball, okay.

01:54 So we have tension, which is at 60 degrees.

02:01 Okay.

02:01 And then we have the normal force.

02:04 So this normal force is not entirely 90 degrees to the tension.

02:09 So it looks like this and if you draw out, so this angle is actually data.

02:19 Okay, and then we have four g, okay, four times the acceleration due to gravity...

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