Homework H4.E - Sp24

Problem statement
Summary sheet for Newton's laws
Solution video

DISCUSSION THREAD

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Discussion

FOUR-STEP PLAN

Step 1: FBD - Draw a free body diagram of particle P. Note that the slot is smooth (no friction), and that the system moves in a horizontal plane (no influence of gravity).

Step 2: Newton - Recommended that you use a set of xy-coordinate axes attached to slotted arm. Resolve your forces into xy-components, and write down Newton's 2nd law for P in terms of its xy-components.

Step 3: Kinematics - It is recommended that you use the moving reference frame velocity and acceleration equations for point P, with the observer being attached to the slotted arm:
vP = vO + (vP/O)rel + ω x rP/O
 aP = aO + (aP/O)rel + α x rP/O + 2ω x (vP/O)rel + ω x (ω x rP/O)
Along with the rigid body velocity and acceleration equations for link AP, you will be able to solve these equations for the xy-components of the acceleration of P.

Step 4: Solve - Combine your equations from Steps 2 and 3 to solve for the normal force acting on P by the slot and the force on P by link AP.

Homework H4.F - Sp24

Problem statement
Summary sheet for Newton's laws
Solution video

DISCUSSION THREAD

Any questions??


As P moves around on the circular track, two things occur:

  1. The normal force N on P due to the circular guide is proportional to the centripetal acceleration  of P: N = mv2/R.
  2. A friction force opposes its motion, where the sliding friction force is proportional to the normal force between the circular guide and P: f = μkN = mμkv2/R.

From this, we see that the friction force goes to zero as the speed goes to zero. What does this imply about P coming to rest? Can you see this in the animation of the motion below?

HINTS:
You will need to use the chain rule of differentiation to set up this problem: dv/dt = (dv/ds)(ds/dt) = v (dv/ds).

Homework H4.C - Sp24

Problem statement
Summary sheet for Newton's Law - 2
Solution video


DISCUSSION THREAD

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Discussion

FOUR-STEP PLAN

Step 1: FBD - Draw individual free body diagrams of A and B, along with an FBD of pulley C.

Step 2: Newton - From each FBD, write down the Newton's equation for components along the incline. Recall that the pulley has negligible mass.

Step 3: Kinematics - You will need to use the cable-pulley system kinematics that we worked with earlier in the semester. Please review the material from Section 1.D of the lecture book to relate the accelerations of blocks A and B.

Step 4: Solve - Combine your equations from Steps 2 and 3 to solve for the accelerations of blocks A and B.

Homework H4.D - Sp24

Problem statement
Summary sheet for Newton's Laws - 2
Solution video


DISCUSSION THREAD

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In the animation of the simulation shown below, the RED vectors shown are the forces of reaction acting on particles A and B (such as the force on each particle by member AB, and the normal forces of reaction by the floor and wall).

Homework H4.A - Sp24

Problem statement
Summary sheet for Newton's Laws - 1
Solution video


DISCUSSION THREAD

Since the motion of P is being described here in terms of polar variables of r and θ, it is recommended that you use a polar description for your forces and acceleration.

Use the Four-Step solution plan outlined in the lecture book:

Step 1 - FBD: Draw a free body diagram of C. NOTE: The arm rotates about a vertical axis, meaning that the arm moves in a horizontal plane; that is, the gravitational force acts perpendicular to the plane of the paper.

Step 2 - Kinetics (Newton): Resolve the forces in your FBD into their polar components. Sum forces in the r-direction and set equal m*ar. Sum forces in the θ-direction and set equal to m*aθ

Step 3 - Kinematics: Use the polar kinematics descriptions of ar = r_ddot - r*θ_dot^2 and aθ = r*θ_ddot + 2*r_dot*θ_dot.

Step 4 - Solve. When solving for the normal force, N, acting on C take note of the sign on your answer. What does this sign mean in terms of answering Part (c)?


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Homework H3.I - Sp24

Problem statement
Summary sheet for 3D MRF kinematics-2
Solution video


DISCUSSION THREAD

Any questions??


Discussion and hints:

Your first decision on this problem is to choose your observer. Since an observer on the plate will have the simplest view of the motion of the insect, attaching the observer to the plate is recommended. Also, attach your xyz-axes to the plate.

Next write down the angular velocity and angular acceleration of the plate. Based on what we have been doing up to this point in Chapter 3, hopefully it is clear that the plate (and observer) has two components of angular velocity: Ω about the fixed X-axis and θ_dot about the moving z-axis. Take a time derivative of the angular velocity vector to find the angular acceleration of the plate (observer).

Following that, determine the motion of the insect as seen by the observer on the plate.

Use these results with the moving reference frame kinematics equation to determine the velocity and acceleration of the insect.