HINTS FOR PROBLEM SET #19

 

Problem Set #19 – Problem 1

Draw the FBD of the rod AB, including the wheels. 
The forces will include the weight, the given force P and the reactions at A and B.

 

Set the FBD equal to a diagram showing the effective force (= mass x acceleration vector) acting at G and the effective moment (= I x alpha).  (You will need to find where G is.)

 

Note that:

1)      the horizontal reaction forces are zero (frictionless)

2)      the vertical reaction at wheel A is zero (given)

3)      alpha is zero (the body is constrained to translate only)

 

Now get the x component of the equation: F = ma

Now get the y component of the equation: F = ma

Now get the k component of the equation: (Sum of the moments about G) = I x alpha

 

These three equations will allow you to solve for the three unknowns,

aX, P and the reaction at B.

 

 
 

Problem Set #19 – Problem 2 

Draw the FBD of the disk. 
The forces will include the tensions in the links BD and AC, and the weight of the disk.

 

Set the FBD equal to a diagram showing the effective force (= mass x acceleration vector) acting at G and the effective moment (I x alpha). 

 

Note that

1)      alpha is zero (the disk is constrained to translate only)

2)      the acceleration of G can be shown by a tangential and a normal component

3)      the normal component is zero (because the disk is released from rest, so the angular velocity is zero)

 

Now get the tangential component of the equation: F = ma

 

Now get the k component of the equation: (Sum of the moments about A) = (Sum of the effective moments about A)  This will include (I x alpha) but will also have another term.

 

Now get the k component of the equation: (Sum of the moments about B) = (Sum of the effective moments about B)  This will include (I x alpha) but will also have another term.

 

These three equations will allow you to solve for the three unknowns: the tension in link BD, the tension in link AC and the tangential component of the acceleration.

 
 

Problem Set #19 – Problem 3

Draw the FBD of the disk. 
The only forces will be one couple from the kinetic friction in the bearings. 

 

Set the FBD equal to a diagram showing the effective force (= mass x acceleration vector) acting at G and the effective moment (I x alpha).  However, note that the acceleration vector is zero! (This body is in rotation only.)

 

Next, you need to get I, the moment of inertia about G.

Next, you need to get alpha.  You can do this from the given information and using the equation for uniform rotational motion.

 

Now plug these into the equation: (Sum of the moments about G) = (I x alpha)

 

 
 

Problem Set #19 – Problem 4 

First we need to get the acceleration of point G.  Then we can use the equation that relates the accelerations of two points on a rigid body to find the absolute acceleration of points A and B:

              aA = aG +  aA/G

              aB = aG +  aB/G

 

To get the acceleration of point G, proceed as in the first 3 problems of this set.  Draw the FBD of rod AB. The forces will include the weight and the reaction of the horizontal surface (neither of which is acting in the plane of motion) and the force P.

 

Set the FBD equal to a diagram showing the effective force (= mass x acceleration vector) acting at G and the effective moment (I x alpha). 

 

Note that the effective force (mass x acceleration vector) has no y component, but does have an x and a z component.

 

Now get the x component of the equation: F = ma

Now get the z component of the equation: F = ma

Now get the k component of the equation: (Sum of the moments about G) = (I x alpha)

 

These three equations will allow you to solve for the acceleration of point G and the angular acceleration of the body, alpha. 

 

Now we will use

              aA = aG +  aA/G

              aB = aG +  aB/G

to get the accelerations of points A and B.

 

The terms aA/G and aB/G will just be the acceleration of points in rotation about a fixed axis.  The tangential components will be (alpha x r) and the normal components will be zero.  (Since it starts from rest.)