A puck of mass m1 is tied to a string and allowed to revolve...

A puck of mass $m_{1}$ is tied to a string and allowed to revolve in a circle of radius $R$ on a frictionless, horizontal table. The other end of table. The other end of the string passes through a small hole in the cen-object of mass $m_{2}$ is tied to it (Fig. P $6.54 ) .$ The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck. (d) Qualitatively describe what will happen in the motion of the puck if the value of $m_{2}$ is increased by placing a small additional load on the puck. (e) Qualitatively describe what will happen in the motion of the puck if the value of $m_{2}$ instead decreased by removing a part of the hanging load.

Key Concepts

Qualitative Analysis of System Variations

This involves understanding how changes in system parameters, such as adding to or reducing the hanging mass, affect the overall dynamics. The analysis requires reasoning about variations in the tension force and the resulting impact on the puck’s speed and circular motion, illustrating the interdependence of the system's parts.

Static Equilibrium

Static equilibrium refers to the state where an object's net force is zero. In this scenario, the hanging mass is in equilibrium, meaning its weight is exactly balanced by the tension in the string. This equilibrium condition is used to link the gravitational force acting on the suspended mass with the tension that also acts on the moving puck.

Tension in a Rope

Tension is the pulling force transmitted along a string or rope. It plays a dual role in this problem by providing the centripetal force required for the puck’s motion and by balancing the weight of the hanging mass. Understanding how to compute tension from both these perspectives is critical.

Newton's Laws of Motion

These laws, particularly the second law (F = ma), provide the foundation for analyzing the forces acting on objects. They are used here to relate the tension in the string to the centripetal force needed for the puck’s circular motion and to understand the balance of forces acting on the hanging mass in equilibrium.

Circular Motion and Centripetal Force

When an object moves in a circle, a net inward (centripetal) force is required to keep it moving along the curved path. In this context, the tension in the string supplies this necessary force for the puck, and the relationship between the speed, mass, and radius of the circle is essential for determining the dynamics of the motion.

Transcript

00:01 Okay, so we have a puck and it's tied to a string and it's revolving in a circle and a radius r and the mass is m1 and then it's on a table and then it's pulled through a hoop and then there's an object of mass two on the other end.

00:25 So i'm not sure the best way to draw that.

00:27 I'll just draw it like this.

00:28 This just means it's going, the dashed means it's going vertically.

00:35 So we want to get, and we know that the radius is not changing.

00:41 And so we want to get the tension and the radial force acting on the puck.

00:46 Okay, so let's just do our usual approach, free body diagrams all around on each mass.

00:58 So for mass one, we can say that we basically just have the tension force acting, and we know, and then we can imply that the sum of the force, at least in the horizontal direction, the vertical direction you have a normal and a weight and those cancel.

01:14 So you can say the sum of the forces is mv squared over r since it's moving in a circle and the force is the tension force.

01:23 And then this applies to m1.

01:29 So we can leave that there.

01:32 And then for i kind of put a little box on it.

01:37 So for two, we have a very similar equation.

01:40 It's not moving up and down.

01:43 So this direction's up, and the tension's holding it up, and then the weight is pulling it down.

01:52 And then, so you can say that the, now this is not actually moving in a circle.

01:58 So we just have the sum of the equation, tension is equal to weight.

02:02 Weight is m2g.

02:05 So, yeah, we can just take this.

02:07 And plug this in here and get some relationships.