Torque and Center of Mass Handout HW #6
Center of Mass:
The center of mass (or mass center) is the mean location of all the mass in a system.
The motion of an object can be characterized by this point in space. All the mass of the object can be thought of being concentrated at this location. The motion of this point matches the motion of a point particle.
Finding the Center of Mass:
Uniform geometric figures have the center of mass located at the geometric center of the object.
Note that the center of mass does not have to be contained inside the volume of the object.
Collections of Point Masses:
The center of mass for a collection of point masses is the weighted average of the position of the objects in space.
Each object will have a position in space. The center of mass is found as:
321
332211
mmm
xmxmxmxcm
321
332211
mmm
ymymymycm
Example #1: A 10.0 kg mass sits at the origin, and a 30.0 kg mass rests at the 12.0 m mark on the x – axis. (a) Find the center of mass for this system.
01 x
kgm 0.101
mx 0.122
kgm 0.302
321
332211
mmm
xmxmxmxcm
kgkg
mkgkgxcm 0.300.10
0.120.3000.10
m00.9
(b) Find the center of mass for this system relative to the mass at the right.
mx 0.121
kgm 0.101
02 x
kgm 0.302
321
332211
mmm
xmxmxmxcm
kgkg
kgmkgxcm 0.300.10
00.300.120.10
m00.3
Although numerically different, it is the same point in space relative to the masses…
Example #2: A 10.0 cm long wire has a mass of 4.00 grams. This wire is bent into an “L” shape that measures 6.00 cm by 4.00 cm, as shown below. Determine the center of mass for this object.
Example #2: A 10.0 cm long wire has a mass of 4.00 grams. This wire is bent into an “L” shape that measures 6.00 cm by 4.00 cm, as shown below. Determine the center of mass for this object.
Treat as two objects:
6 cm object:
gcmcm
gm 40.200.6
0.10
00.41
0,1 cmx
cmy cm 00.3,1
4 cm object:
gcmcm
gm 60.100.4
0.10
00.42
cmx cm 00.2,2 0,2 cmy
321
332211
mmm
xmxmxmxcm
g
cmggxcm 00.4
00.260.1040.2 cm800.0
321
332211
mmm
ymymymycm
g
gcmgycm 00.4
060.100.340.2 cm80.1
Example #3: Determine the center of mass of the following masses, as measured from the left end. Assume the blocks are of the same density.
Torque
Torque is the rotational equivalent of force. A torque is the result of a force applied to an object that tries to make the object rotate about some pivot point.
Equation of Torque:
pivot point
r distance from pivot to applied force
F
applied force
angle between direction of force and pivot distance.
sinrFtorque
Note that torque is maximum when the angle is 90º.
The units of torque are Nm or newton · meter
The torque is also the product of the distance from the pivot times the component of the force perpendicular to the distance from the pivot.
rFrFtorque sin
Example #5: Calculate the total torque about point O on the figure below. Take counterclockwise torques to be positive, and clockwise torques to be negative.
sinrF
20sin100.460sin250.2 NmNmnet
Nmnet 6.29
Example #6: The forces applied to the cylinder below are F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, and F4 = 5.0 N. Also, R1 = 5.0 cm and R2 = 12 cm. Determine the net torque on the cylinder.
90sin21 RFnet
90sin22 RF
90sin13 RF
0sin24 RF0
mNmNmNnet 050.00.212.00.412.00.6
Nmnet 14.0