Group Assignment- Equilibrium/Rolling
Due: 12-8-10
Answer the following questions, showing all work for credit. Be sure to include appropriate diagrams, equations, and substitutions. Working in groups is encouraged but the work is handed in individually.
Read:
Example Problem on back of this sheet
Chapter 12 sections 1-5, page 314 gives excellent Problem-Solving Tactics
Look at Chapter 10 Rotation Notes 23-24
Look at Chapter 11 Rolling Notes 15-27
Complete/Hand in:
Worksheet “Equilibrium/Rolling Group Assignment” =160 pts
Example: Equilibrium
If the Net force =0 and Net torque=0, then it is in equilibrium. Equilibrium could mean that the object is at rest or at a constant velocity.
If it is not rotating then where is the axis of rotation?
You pick the axis of rotation, usually at an unknown force giving it a radius of zero. When a force has a radius of zero, then its torque is zero and it is no longer in the equation.
A 50.0 kg uniform square sign, 2.00 m on a side, is hung from a 3.00 m horizontal rod of negligible mass. A cable is attached to the end of the rod and to a point on the wall 4.00 m above the point where the rod is hinged to the wall. What is the tension in the cable?
q
Put axis of rotation at Hinge so no hinge forces appear in the torque equation
Torque of Hinge (zero) + Torque of tension- torque of weight of sign = net torque (zero)
H (0) + Tr - mgr=0
T(3)sin q - (50g)2=0
Angle: tan q=4/3 q=53.1
T (3) sin 53.1-50g2=0
T=408.5 N
a. What is the magnitude and direction of the horizontal components of the force on the rod from the wall?
X force equation (NO TORQUES, just forces in the x-direction)
Fhinge-Tcosq=0
Fhinge=408.5 cos 53.1=245 N
b. What is the magnitude and direction of the vertical components of the force on the rod from the wall?
Y: equation (NO TORQUES, just forces in the y-direction)
Fhinge-mg +Tsinq=0
Fhinge=50g -408.5sin53.1
Fhinge=163.7N ( Positive answer which indicates force of hinge is up!)
Equilibrium/Rolling Group Assignment
Name______
Score______/160 pts
1. (problem 3) A rope of negligible mass is stretched horizontally between two supports that are 3.44 m apart. When an object of weight 31600 N is hung at the center of the rope, the rope is observed to sag by 35 cm.
- Draw a freebody diagram and write equations (No torques necessary here, just do forces and Newton’s 2nd Law).
- Find the tension in the rope.
2. Page 322 problem 12
- Draw a freebody diagram (for the two “knots”) and write relevant equations.
- Find tension T1, T2, T3, θ.
3. A 350 N uniform boom is hinged to a vertical wall held horizontal by a cable as shown below. The boom has a length 4.0 m, and a cable is attached at 2/3L from the wall. A mass of 50.0 kg hangs at the end of the boom.
- Sketch below the forces acting on the boom.
- Find the Tension in the cable (use Torque equation).
- Determine the horizontal and vertical components of the force of the hinge on the boom (Use vertical/horizontal force equations).
4. See Sample Problem 12-1 page 310. A diver of weight 580 N stands at the end of a 4.5 m diving board of negligible mass. The board is attached to two pedestals 1.5 m apart. What are the magnitude and direction of the force on the board from (a) the left pedestal?
b. What are the magnitude and direction of the force on the board from the right pedestal?
c. Which pedestal is being stretched? Which pedestal is being compressed?
Problem 27 in book see figure 12-43
6. The system above is in equilibrium. A concrete block of mass 225 kg hangs from the end of the uniform strut whose mass is 45 kg. (see sample problem 12-3, remember that only a force perpendicular to the radius produces a torque on the object…. Weight is not perpendicular here)
- Find the tension T in the cable
- Find the horizontal force component on the strut from the hinge
- Find the vertical force component on the strut from the hinge.
7. Page 323 problem 25. (15pts)
8. Page 324 problem 30 (15pts)
9. Page 320 Question 3
See Sample Problem 12-2 on page 311. A uniform ladder of weight W leans without slipping against a wall making an angle θ with a floor as shown above. There is friction between the ladder and the floor, but the friction between the ladder and the wall is negligible.
10. The magnitude of the normal force exerted by the floor on the ladder is
a) W
b) Wsinθ
c) Wcosθ
d) W/2 sinθ
e) W/2 cosθ
11. The magnitude of the friction force exerted on the ladder by the floor is
a) 2Wtan θ
b) W
c) Wcotθ
d) W/2
e) W/2 cotθ
Rolling Forward
12. A thin hoop of mass M, radius R. and rotational inertia MR2 is released from rest from the top of the ramp of length L above. The ramp makes an angle θ with respect to a horizontal tabletop to which the ramp is fixed. The table is a height H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Express all algebraic answers in terms of given quantities and fundamental constants.
a. On the figure below, draw and label the forces acting on the hoop as it rolls down the inclined plane Your arrow should begin at the point of application of each force.
b. Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp.
c. Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp.
d. Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor.
e. Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined in part (c) for the hoop'?
_____Less than ______The same as _____Greater than
f. Determine the minimum coefficient of friction between the hoop and the inclined plane that is required for the hoop to roll without slipping.
g. The coefficient of friction µ is now made less than the value determined in part (f), so that the hoop both rotates and slips.
i. Indicate whether the translational speed of the hoop at the bottom of the inclined plane is greater than, less than, or equal to the translational speed calculated in part (c). Justify your answer.
ii. Indicate whether the total kinetic energy of the hoop at the bottom of the inclined plane is greater than, less than, or equal to the total kinetic energy for the previous case of rolling without slipping. Justify your answer.
14. The cart shown above is made of a block of mass m and four solid rubber tires each of mass m/4 and radius r. Each tire may be considered to be a disk. (A disk has rotational inertia ½ ML2, where M is the mass and L is the radius of the disk.) The cart is released from rest and rolls without slipping from the top of an inclined plane of height h. Express all algebraic answers in terms of the given quantities and fundamental constants.
(a) Determine the total rotational inertia of all four tires.
(b) Determine the speed of the cart when it reaches the bottom of the incline.
(c) After rolling down the incline and across the horizontal surface, the cart collides with a bumper of negligible mass attached to an ideal spring, which has a spring constant k. Determine the distance xm the spring is compressed before the cart and bumper come to rest.
(d) Now assume that the bumper has a non-neglible mass. After the collision with the bumper, the spring is compressed to a maximum distance of about 90% of the value of xm in part (c). Give a reasonable explanation for this decrease