# Dynamic Final Assignment – Mechanical Engineering

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1. An elliptical rigid body has a mass of 25kg and a radius of gyration of 0.4 m. It is initially rotating clockwise about Point A at 3 rad/s. The net counterclockwise moment about Point A is given as a function of time on the graph below. What is the speed and direction of rotation of the object at time = 3.5 seconds?

time [s] M [N m] 2.0 3.5 20 !” = 3 rad/s%(‘) A m = 25 kg k = 0.4 m !”#\$% \$&&”‘#()#*

1. A disk has a mass of 30kg and a radius of 0.4m. It is rolling without slipping on a stationary cylinder with a radius of 1 m. When it is 30° past vertical, its angular velocity is 5 rad/s. What is its angular acceleration, , at that position?

G ! = 5 rad/s \$ = ? ‘ = 0.4 + ! 30 ° Disk m = 30 kg G ! = 5 rad/s \$ = ? ‘ = 0.4 + ! 30 ° Disk m = 30 kg N = ? rolling without slipping

1. For the situation in Problem 2, what is the normal force, N, that the cylinder is putting on the disk?
2. A rigid object has a mass of 18kg and a radius of gyration of 0.3m. It is initially rotating clockwise about a fixed axis at its center of gravity with an angular speed of 25 rad/s. There is a constant friction moment of 5 Nm. How many revolutions will the object make before it stops rotating?

!” = 25 ‘()/+,- = 5 . / m = 18 kg k = 0.3 m

1. An uniform isosceles triangle has a mass of 15 kg, a height of 0.33m and a base of 0.26m. It is initially sliding to the left on a smooth horizontal surface at 2 m/s (State 1). It hits a small stop, and does not bounce off of the stop. What is the angular speed, &, just after the triangle hits the stop (State 2)?

0.26 m 0. 33 m m = 15 kg !” = 2 m/s ! State 1 Just before triangle hits stop ! State 2 Just after triangle hits stop !” = ?

1. A uniform rectangular block rotates with negligible friction about a fixed axis at its corner (point A). A horizontal force of 130 N is applied to the block when it is in the position shown. What is the direction and magnitude of its angular acceleration, , at this time?

F = 130 N A ! 8 m 0.6 m ! = ? 24 kg

1. A moving pulley can be treated as a uniform disk with a mass of 100 kg and a radius of 0.8 m. It is being accelerated upward by a cable that is attached to a stationary point on one side (right) and an applied force, F, on the other side (left). The center of the pulley is accelerating upward at 2 m/&. What is the magnitude of the applied force, F?

stationary F = ? !” ! Disk m = 100 kg R = 0.8 m !” = 2 m/&’

1. A rigid object has a uniform area density of 100 kg/&. It consists of a 1.2m by 1.2m square with four 0.2m radius holes cut out. The holes are tangent to the outside of the square and at the midpoint of each side. What is the mass moment of inertia of the object about Point A, which is at the center of the square (object is rotating in the plane of the page)?

A 1.2 m 2 m r = 0.2 m 100 kg/!”

1. A uniform rectangular block is attached to a cart that is accelerating up a 20° incline at 3 m/&. The block is attached to a cart by a hinge at Point A and a roller at Point B. What is the force that the roller is applying to the block at Point B?

20° \$m= 1 00 kg 8 m 0.6 m 0 m ! = 3 m/\$ % X Y A B 15 kg !” = 4.8 m/s stationary ! !’ = ? *+ = 0.8 – *. = 0.3 -10 kg

1. A block is attached to a moving spool. A cable wraps around the outside of the spool, which is attached to a stationary point on the right side, and moving upward at 4.8 m/s on the left side. A 15kg block is attached by a cable to the inner radius of the spool. What is the magnitude and direction of the velocity of the block.
2. A small bi-plane is doing a vertical loop at an airshow. When it is at the bottom of the loop, it has a constant speed of 35 m/s, and the radius of curvature of its path is 100m. When looking from the front of the plane, the propeller is spinning clockwise with a constant speed of 160 rad/s. What is the magnitude and direction (relative to the plane) of the angular acceleration vector, �⃗�, of the propeller?

!” = 160 ()*/, FRONT VIEW – ! = 100 & ‘( = 35 &/, – SIDE VIEW ! !!! !

1. A 2kg hoop, with a radius of 0.6m, is released with backspin onto a 25° incline. It is released smoothly, so it does not bounce. The kinetic coefficient of friction between the hoop and the surface is 0.4 and the initial speed of the center of the hoop is 2 m/s up slope and the initial angular speed of the hoop is 3 rad/s clockwise. What are the initial linear acceleration of the center of the hoop and initial angular acceleration of the hoop (magnitudes and directions)?

!” = 3 rad/s 25°) Ring m = 2 kg R = 0.6 m ! = 3 rad/%& B30°)* = 4 m/%&, = 2 rad/s Y X.* = 6 m/s)⃗1 = ?

1. A tram is moving to the left at 6 m/s and decelerating at 4 m/&. A 2m rotating rigid rod is attached to the bottom of the tram. The rod is rotating counterclockwise with an angular velocity of 2 rad/s and its angular speed is decreasing at 3 rad/&. At that time the rod is oriented 30° counterclockwise from the negative y-axis (as shown). What is the absolute acceleration of the end of the rod, Point B?
2. A cable is attached to the center of a uniform disk on one end and a block on the other, as shown. At State 1, the block and the disk are stationary. The disk rolls without slipping between State 1 and State 2. Ignore mass of the cable and the small stationary pulley, and ignore friction losses in the system. What is the downward speed, -&, of the block after it has dropped 2m?

B Block B ‘( = 50 kg – Disk A ‘2 = 80 4- 52 = 0.5 ‘rolling without slipping A 7(8 = 0 98 = 0 State 1 ! B ! rolling without slipping A “#\$ = ?

State 2 ∆) = * + ,\$ !

Mass Moments of Inertia for some simple shapes about an axis going through their center of gravity (perpendicular to the paper): Sphere: Circular Disk: Thin Ring: !” = \$ % &’ \$ !” = ( \$ &’ \$ !” = &’\$ Rectangular Plate: Slender Rod: !” = ( (\$ &(* \$ + ,\$) !” = ( (\$ &. \$

Isosceles Triangle: Equilateral Triangle:

!” = & 01 \$2 + 31 (4 !” = ( (\$ &. \$ L a b G G h/3 b h b/2 G L L L

Last Updated on June 9, 2021