Tutorial Sheet 4: Planar Dynamics
Topics covered are
- Forces, moments, and inertia
- SUVAT
- Inertia from different axes
Tips
- With lots of directions and taking forces from different locations, make sure you don’t get mixed up.
- aka please draw them out trust me and mark on where your origin is.
- I include circle arrows to indicate which direction the moments are taken around.
- Sometimes part a and b have to be solved in different orders.
Question 1
A horizontal force F=133.4N is applied to the 1023N refrigerator as shown, and the fridge does not tip over. Friction is negligible.
(a) What is the magnitude of the refrigerator’s acceleration?
(b) What normal forces are exerted on the refrigerator by the floor at A and B?
Answer
(a)
\[a= 1.27\](b)
\[A = 359N, B=664N\]Question 2
Solve Q1 if the coefficient of kinetic friction at A and B is $\mu_k$=0.1.
Answer
(a)
\[a=0.3\](b)
\[A = 256.7N, B=766.3N\]Question 3
As the 2800 N airplane begins its take off run at t=0, its propeller exerts a horizontal force T=1000 N. Neglect horizontal forces exerted on the wheels by the runway.
(a) What distance has the airplane moved at t=2s?
(b) What normal forces are exerted on the tires at A and B?
Answer
(a)
\[s = 7 m\](b)
\[A=942.9N, B=1857N\]Question 4
The crane moves to the right with constant acceleration, and the 800-kg load moves without swinging.
(a) What is the acceleration of the crane and load?
(b) What are the tensions in the cables attached at A and B?
Answer
(a)
\[a = 0.858 \text{ m/s}^2\](b)
\[A = 3709N, B=4169N\]Question 5
The moment of inertia of the disk about O is I=20 kgm $^2$.At t=0, the stationary disk is subjected to a constant 50 Nm torque.
(a) What is the magnitude of the resulting angular acceleration of the disk?
(b) How fast is the disk rotating (in rpm) at t=4s?
Answer
(a)
\[\alpha=2.5 \text{ rad/s}^2\](b)
\[95.5 \text{ rpm}\]Question 6
The radius of the pulley is 125 mm and the moment of inertia about its axis is I=0.05 kgm $^2$. If the system is released from rest, how far does the 20 kg mass fall in 0.5 s? What is the tension in the rope between the 20 kg mass and the pulley?
Answer
\(s = 0.721m\) \(80.8N \text{ tension in rope }, \alpha=46.2 rad/s^2\)
Question 7
Points B and C lie in the x–y plane. They axis is vertical. The center of mass of the 18 kg arm BC is at the midpoint of the line from B to C, and the moment of inertia of the arm about the axis through the center of mass that is parallel to the z axis is 1.5 kgm $^2$. At the instant shown, the angular velocity and angular acceleration vectors of arm AB are $\omega_{AB}$=0.6k rad/s and $\alpha_{AB}$=−0.3k rad/s $^2$.The angular velocity and angular acceleration vectors of arm BC are $\omega_{BC}$=0.4k rad/s and $\alpha_{BC}$=2k rad/s $^2$. Determine the force and couple exerted on arm BC at B.
Answer
Then use dynamic equations
\[B_x=-19.1N, B_y = 183.3N\] \[M_B=62.6Nm\]Question 8
An engineer gathering data for the design of a maneuvering unit determines that the astronaut’s center of mass is at x=1.01 m, y=0.16 m and that her moment of inertia about the z axis is 105.6 kgm $^2$. The astronaut’s mass is 81.6 kg. What is her moment of inertia about the z’ axis through her center of mass?
Answer
\[20.27 kgm^2\]Question 9
The mass of the object is 10 kg. Its moment of inertia about L $_1$ is 10 kgm $^2$. What is its moment of inertia about L $_2$? The three axes are in the same plane.
Answer
We can first determine MoI around L, which can be then used to find it around L $_2$
\[20.8 kgm^2\]Question 10
Two homogenous slender bars, each of mass m and length l, are welded together to form the T-shaped object. Use the parallel-axis theorem to determine the moment of inertia of the object about the axis
(a) through point O that is perpendicular to the plane formed by the bars. (b) through the center of mass of the object that is perpendicular to the plane formed by the bars.
Answer
(a)
\[= \frac{17}{12}ml^2\](b)
\[\frac{7}{24}ml^2\]Question 11
The rocket is used for atmospheric research. Its weight and its moment of inertia about the z axis through its center of mass (including its fuel) are 44480 N and 13826 kgm $^2$, respectively. The rocket’s fuel weighs 26688 N, its center of mass is located at x=−0.91, y=0, and z=0 m,and the moment of inertia of the fuel about the axis through the fuel’s center of mass parallel to z axis is 2983 kgm $^2$. When the fuel is exhausted, what is the rocket’s moment of inertia about the axis through its new center of mass parallel to z axis?
Answer
\[5186 kgm^2\]Question 12
Model the arm ABC as a single rigid body. Its mass is 320 kg, and the moment of inertia about its center of mass is I=360 kgm $^2$. Point A is stationary. If the hydraulic piston exerts a 14 kN force on the arm at B what is the arm’s angular acceleration?
Answer
\[0.59 rad/s^2 \text{ anticlockwise}\]Question 13
A thin ring and a homogeneous circular disk, each of mass m and radius R, are released from rest on an inclined surface. Determine the ratio $v_{ring}/v_{disk}$ of the velocities of the their centers when they have rolled a distance D.
Answer
\(\sqrt{\frac{3}{4}}\)