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Chapter 12: Problem 4

Why is a half-filled tumbler more stable than an empty one or a completely filled one?

Short Answer

Expert verified
A half-filled tumbler is more stable than an empty one or a fully filled one because its center of mass is in the middle, providing a balance between the high instability of a top-heavy filled glass and the light weight of an empty glass. Thus, it has a good combination of weight and a low enough center of mass to make it more stable against being knocked over.

Step by step solution

01

Understanding Center of Mass

The center of mass is the point in an object where all of its weight can be considered to be concentrated. In the case of the tumbler, when it is empty, the Center of Mass (COM) lies at its geometrical center. When filled with liquid, the COM rises as the mass increases at the top.
02

Understanding Torque

Torque is the rotational equivalent of linear force. The amount of torque depends on how much force is being applied, and the distance from the center of rotation(SE). A tumblers stability is greater when its center of mass is closer to the surface it鈥檚 standing on. A filled tumbler has a high center of mass and will require only a small amount of sideways force to knock over. An empty glass, being lighter, can be easily knocked over.
03

Importance of Being Half-Filled

When the tumbler is half-filled, the center of mass is lower than when it is fully filled, but higher than when it is empty. This 'medium' position of the center of mass provides more stability than the other two conditions. Because it is partially filled, it also has more weight than an empty one which gives it resistance against being knocked over.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Torque in Everyday Objects
Torque is an interesting concept that plays a role in how objects move and stay balanced. Imagine opening a door. You apply force at a handle to swing it open or closed. That's torque in action! The further your hand is from the hinges, the easier it is to open the door because you're applying more torque with the same force.

Torque is vital in understanding how objects like tumblers remain stable or topple over. Stability depends on how the force acts relative to the object's center of mass. The formula for torque is:
  • Torque = Force \( \times \) Distance from pivot point
When you push a tumbler, the stability is determined by how far the force you've applied can rotate it up and over its edge. If it's full, just a little nudge can cause it to fall because the center of mass is higher.
Understanding Tumbler Stability
Stability of objects, like half-filled tumblers, depends on the location of the center of mass. Picture a trapeze artist balancing on a tightrope: they need their center of mass directly above their support to maintain balance. The same principle applies to tumblers and other household objects.

Here's what's happening with that tumbler:
  • Empty Tumbler: The low center of mass (COM) makes it easy to tip with a gentle push.
  • Half-filled Tumbler: The weight of the liquid lowers the COM compared to a full tumbler but still provides resistance to tipping due to the added mass.
  • Full Tumbler: A high COM makes it unstable and prone to toppling with minor disturbances.
A half-filled tumbler strikes a balance: the COM is low enough to be stable but heavy enough to resist small forces.
Dive into Rotational Dynamics
Rotational dynamics is the branch of physics that studies how forces affect rotational motion. It's key in predicting how and why objects spin or topple over.

In the case of the tumbler exercise, rotational dynamics help us to understand the stability through the concepts of:
  • Moment of Inertia: Determines how the mass is distributed around the axis of rotation. More mass far from the axis increases inertia and stability.
  • Angular Velocity: Though not directly related to stability, it refers to how fast an object rotates.
Consider how fast a top spins smoothly if its mass is evenly distributed. The same logic applies to tumblers as their stability changes. The half-filled condition offers a favored moment of inertia that contributes to its stable state.

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Most popular questions from this chapter

Figure \(12.18 a\) shows an outstretched arm with mass \(4.2 \mathrm{~kg}\). The arm is \(56 \mathrm{~cm}\) long, and its center of gravity is \(21 \mathrm{~cm}\) from the shoulder. The hand at the end of the arm holds a \(6.0\)-kg mass. (a) Find the torque about the shoulder due to the weight of the arm and the \(6.0-\mathrm{kg}\) mass. (b) If the arm is held in equilibrium by the deltoid muscle, whose force on the arm acts at \(5^{\circ}\) below the horizontal at a point \(18 \mathrm{~cm}\) from the shoulder joint (Fig. 12.18b), what's the force exerted by the muscle?

The best way to lift a heavy weight is to squat with your back vertical, rather than to lean over. Why?

When the boom rope is horizontal, it can't exert any vertical force. Therefore, a. it's impossible to hold the boom with the boom rope horizontal. b. the boom rope tension becomes infinite. c. the pivot supplies the necessary vertical force. d. the boom rope exerts no torque.

A uniform \(5.0-\mathrm{kg}\) ladder is leaning against a frictionless vertical wall, with which it makes a \(15^{\circ}\) angle. The coefficient of friction between ladder and ground is \(0.26\). Can a \(65-\mathrm{kg}\) person climb to the top of the ladder without it slipping? If not, how high can that person climb? If so, how massive a person would make the ladder slip?

A portion of a roller-coaster track is described by the equation \(h=0.65 x-1.3 \times 10^{-2} x^{2}\), where \(h\) and \(x\) are the height and horizontal position in meters. (a) Find a point where the rollercoaster car could be in static equilibrium on this track. (b) Is this equilibrium stable or unstable?
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