Please note that since P is not sliding on the rotating guide, P is traveling along a horizontal circular path having a radius with the radius r being the perpendicular distance from P to the vertical shaft. It is recommended that you use a set of polar coordinates: e_r pointing outward from the vertical shaft to O; e_φ tangent to the above-described path of P; and, k pointing upward.

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

Step 1 – FBD: Draw a free body diagram of P. With the guide being smooth, there will be only two forces acting on P: the weight and the normal force N from the rotating guide.

Step 2 – Kinetics (Newton): Resolve the forces in your FBD into their polar components. Sum forces in the r-direction and set equal m*a_r. Sum forces in the k-direction and set equal to 0 (since P has no vertical motion for all time).

Step 3 – Kinematics: Use the polar kinematics descriptions of the acceleration of P. Note that r is constant for all time and Ω is constant.

Step 4 – Solve. With the above equations you will have sufficient number of equations to solve for the unknowns in the problem, which includes N and θ.

Any questions?? Please ask/answer questions regarding this homework problem through the “Leave a Comment” link above.

Any questions??

Discussion and hints:

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

Next write down the angular velocity and angular acceleration of the tube. Based on what we have been doing up to this point in Chapter 3, hopefully it is clear that the tube (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 tube (observer).

Following that, determine the motion of the particle P as seen by the observer on the tube.

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

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.

Any questions??

Discussion and hints:

It is recommended that you use an observer attached to the boom. As we have discussed in class, your choice of observer directly affects four terms in the acceleration equation: ω and α  (how the observer moves), and the relative velocity and relative acceleration terms (what the observer sees). Note that the remainder of the discussion here is based on having the observer attached to the boom.

The boom shown above has TWO components of rotation:

• a rotation rate of Ω about a fixed axis (the “+” Y-axis), and,
• a rotation rate of θ_dot about a moving axis (the “+” z-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the wheel is given by:

ω = Ω J + θ_dot k

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

• Recall that the J-axis is fixed. Since J is fixed, then dJ/dt = 0.
• Recall that the k-axis is NOT fixed. Knowing that, how do you find dk/dt?

With the observer attached to the boom, what motion does the observer see for point P? That is, what are (v_P/O)_rel and (a_P/O)_rel?

NOTE: Pay particular attention to the motion of the reference point O. What path does O follow? And, based on that, how do you write down the acceleration vector of point O, a_O?

Any questions??

Discussion and hints:

It is recommended that you use an observer attached to the wheel. As we have discussed in class, your choice of observer directly affects four terms in the acceleration equation: ω and α  (how the observer moves), and the relative velocity and relative acceleration terms (what the observer sees). Note that the remainder of the discussion here is based on having the observer attached to the wheel.

The wheel shown above has TWO components of rotation:

• a rotation rate of ω₁ about a fixed axis (the “+” Y-axis), and,
• a rotation rate of ω₂ about a moving axis (the “+” z-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the wheel is given by:

ω = ω₁J + ω₂ k

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

• Recall that the J-axis is fixed. Since J is fixed, then dJ/dt = 0.
• Recall that the k-axis is NOT fixed. Knowing that, how do you find dk/dt?

With the observer attached to the wheel, what motion does the observer see for points A and B? That is, what are (v_A/O)_rel and (a_A/O)_rel, and (v_B/O)_rel and (a_B/O)_rel?

NOTE: Pay particular attention to the motion of the reference point O. What path does O follow? And, based on that, how do you write down the acceleration vector of O, a_O?

Any questions??

Discussion and hints:

The disk shown above has TWO components of rotation (note that ω₀ = 0):

• a rotation rate of θ_dot about a fixed axis (the “+” K-axis), and,
• a rotation rate of ω_disk about a moving axis (the “-” x-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the disk is given by:

ω =θ_dot* K – ω_disk i

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

• Recall that the K-axis is fixed. Since K is fixed, then dK/dt = 0.
• Recall that the i-axis is NOT fixed. Knowing that, how do you find di/dt?

Any questions??

Discussion and hints:

It is RECOMMENDED that you choose to put your observer on the disk. In that case, the ω and α  that go into your acceleration equation will be that of the disk.

The discussion below follows a choice of putting the observer on the disk.

The disk shown above has TWO components of rotation:

• a rotation rate of ω₀ about a fixed axis (the “-” Z-axis), and,
• a rotation rate of ω_disk about a moving axis (the “+” y-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the disk is given by:

ω = -ω₀K + ω_disk j

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

• Recall that the K-axis is fixed. Since K is fixed, then dK/dt = 0.
• Recall that the j-axis is NOT fixed. Knowing that, how do you find dj/dt?

Any questions??

Discussion and hints:

Let’s first take a look at the motion of point B. This motion of B is shown in the simulation results below.

The motion of B is circular, with the center of the path located at point D. This is expected since link BD is pinned to ground at point D.

Now, let’s attach an observer to link AC. Keep in mind that this observer is unaware that they are moving. The motion of B that this observer sees is straight, with this straight path aligned with the collar sliding on link AC. Since the xy-axes are attached to AC, this relative motion is in the x-direction:

(v_B/A)_rel = v_rel i = d_dot i

(a_B/A)_rel = a_rel i = d_ddot i

where d_dot and d_ddot are unknowns (you will solve for these in your analysis). You can see this is the following animation shown from the perspective of the observer attached to link AC.

Discussion

• Shown above left is the motion of the mechanism as seen by a fixed observer. Note that the stationary observer sees P moving on a circular path centered at point A.
• And, shown above right is the motion of P as seen by an observer attached to the slotted arm. Note that the moving observer sees P moving on a circular path centered at point C.

It is suggested that you employ the view of the observer on the slotted arm in your analysis.

Any questions?? Please ask/answer questions regarding this homework problem through the “Leave a Comment” link above.

Any questions??

Discussion:

Let’s first take a look at the motion of point D. This motion of D is shown in the simulation results below.

The motion of D is circular, with the center of the path located at point A. This is expected since link AD is pinned to ground at point A.

Now, let’s attach an observer to link BE. Keep in mind that this observer is unaware that they are moving. The motion that this observer sees is straight, with this straight path aligned with the slot cut into link BE. You can see this is the following animation shown from the perspective of the observer attached to link BE.