Ch 01: Units & VectorsWorksheetSee all chapters
All Chapters
Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
Ch 11: Rotational Inertia & Energy
Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

Solution: In an assembly operation illustrated in the figure, a robot moves an object first straight upward and then also to the east around an arc forming one-quarter of a circle of radius 4.80 cm that lies in

Solution: In an assembly operation illustrated in the figure, a robot moves an object first straight upward and then also to the east around an arc forming one-quarter of a circle of radius 4.80 cm that lies in

Problem

In an assembly operation illustrated in the figure, a robot moves an object first straight upward and then also to the east around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an east-west vertical plane. The robot then moves the object upward and to the north through one-quarter of a circle of radius 3.70 cm that lies in a north- south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.

Solution

When adding vectors in 3D, the steps are basically the same as adding vectors in 2D, except that drawing diagrams for 3D problems is often skipped because it's more complicated to do.

  1. Resolve the vectors into components, if necessary.
  2. Organize your known information.
  3. Add or subtract vectors as required.
  4. Convert back to magnitude-angle notation (unless the problem only asks for components).

For step 1, we'll usually use equations of this form to find our components when the components aren't given to us:

Ax=|A|cos α,  Ay=|A|cos β, Az=|A|cos γ

Steps 2 and 3 can be done using a table.

For step 4, we'll use the same equations from step 1 solved for α, plus the equation for magnitude:

|A|=Ax2+Ay2+Az2

This problem looks tricky at first because it's very wordy! But let's tackle it one piece at a time:

Step 1. Notice that the first displacement, let's call it A, is a quarter-circle that starts with a motion straight upwards. The path is along an arc, but the displacement is from the starting point at the edge of the arc to the final point. If we imagine an origin (0,0) at the center of the arc (or sketch it out that way), it's easy to see that the initial position vector is horizontal, the final position vector is vertical, and both of them have magnitudes equal to the radius of the circle (4.80 cm).

Knowing that, we don't have to do any math to find the components: we can just say

Aup=4.80 cm and Aeast=4.80 cm.

We can use the same reasoning for the second displacement, B. We know it's a quarter-circle up and to the north with radius 3.70 cm, so

Bup=3.70 cm and Bnorth=3.70 cm.

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