Mechanics II

Question 1

Equation for finding center of mass

Answer 1

(x₁m₁) + (x₂m₂) + (x₃m₃) … + (X_nm_n)

_________________________________

m₁ + m₂ + m₃ + m_n

Question 2

Is the center of mass to the R or L of the origin?

Answer 2

Depends on where you place your zero mark. Usually you place your zero mark all the way to the L end of the origin. If this is the case, the center of mass would be to the R of that origin (as you would always have a positive number). However if you chose to use the middle of a pole/stick for an origin, a positive number would mean that the center of mass is to the R of that origin, while a negative number would mean that the center of mass is to the L of that origin.

Question 3

An object moving in a circular path is said to execute uniform circular motion if its ____ is constant

Answer 3

speed

Question 4

Which way does velocity, acceleration, and centripetal force point when an object undergoes uniform circular motion?

Answer 4

the velocity vector always points tangent to the objects path, even though the magnitudes of these vectors are always the same (because speed is constant)

centripetal acceleration and centripetal force always point towards the center of the circle

Question 5

Equation for centripetal acceleration (a_c)

Answer 5

a_c = ​v²/r

Question 6

Equation for centripetal force (F_c)

Answer 6

F_c = ma_c = mv² / r

Question 7

Equation for translation velocity (for an object experiencing uniform centripetal motion)

Question 8

Will centripetal acceleration increase or decrease for an object if it is moved further from the axis of rotation?

Answer 8

a_c will increases for an object when placed further from the axis of rotation

translational velocity: v = rw

a_c = v² / r

therefore, a_c = rw² → a_c is proportional to r, so as r increases, there is a proportional increase in a_c

Question 9

If an object undergoing uniform circular motion is being acted upon by a constant force toward the center, why doesn’t the object fall into the center?

Answer 9

Actually, it is falling toward the center, but because of its speed, the object remains in a circular orbit around the center. Remember the direction of v is not necessarily the same as the direction of Fnet. So, just because Fnet points toward the center doesn’t mean that v must point toward the center. It’s the direction of the acceleration, no the velocity that always matches the direction of Fnet. Furthermore, if it weren’t for Fnet point toward the center (that is, if the F_c were suddenly removed), then the object’s velocity wouldn’t change (it would fly off in a straight line tangent to the circle at the point where the force was removed)

Question 10

How would the net force on an object undergoing uniform circular motion have to change if the object’s speed doubled?

Answer 10

F_c would have to increase by a factor of 4

F_c = mv²/r → F_c is proportional to the square of the speed

If speed increases by a factor of 2, F_c would increase by a factor of 2² = 4

Question 11

Equation for force of gravity (F_grav)

Answer 11

F_grav = GMm / r²

Question 12

The moon orbits the eart in a nearly circular path at nearly constant speed. If M is the mass of the earth, m is the mass of the moon, and r is the radius of the moon’s orbit, find an expression for the speed of the moon’s orbit.

Answer 12

First, determine what is providing the centripetal force? The answer is the gravitational pull by the earth. Thus, set F_grav equal to F_c:

GMm/r²=mv²/r → GM/r=v² → v=(GM/r)^0.5

Note that the mass of the moon cancels out; thus the mass of any object orbiting at the same distance from the earth as the moon must move at the same speed as the moon.

Question 13

A rope is tied to the handle of a bucket, and the bucket is then whirled in a vertical circle. At the bottom of its path, the tension of the rope is 50N. How would you find the net force of this bucket?

Answer 13

The net force of the bucket (and therefore the centripetal force of the bucket) would be the tension force minus the weight force:

F_net = F_T - F_w = 50N - mg

m = mass of bucket

Question 14

What’s the difference between centripetal acceleration and tangential acceleration in regards to speed?

Answer 14

centripetal acceleration only makes an object turn so that it moves in a circular path; therefore it doesn’t change the speed.

tangential acceleration changes the speed of the object, as it is the force component (vector) that opposes the direction of the object’s velocity (therefore is the opposite tangent to the velocity tangent; both tangents are perpendicular to the center of the circle, just in opposite directions). This is the reason that an object’s speed decreases when traveling from the bottom to the top of a circle, and increases when traveling from the top to the bottom of a circle

Question 15

Equation for torque

Answer 15

T = rFsin(theta)

or

T = lF (l is the lever arm, which is always perpendicular to the line of action of the Force applied, which means theta is 90º, and sin90º=1)

(T is really the Greek letter tau)

Question 16

At what angle should the force be applied to give the max amount of torque?

Answer 16

90º because sin90º = 1

Question 17

In general, if a force at the pivot (or along a line through the pivot), then the torque is ____

Answer 17

zero

Question 18

As r increases, torque ____ (increases or decreases)

Answer 18

increases

Question 20

A homogenous rectangular sheet of metal lies on a flat table and is able to rotate around an axis through its center, perpendicular to the table. Four forces of equal magnitude are applied, and each one is exerted at each corner of the sheet in a counterclockwise direction. Which one of the following statements is true?

A) The net force is zero, but the net torque is not

B) The net torque is zero, but the net force is not

C) Neither the net force nor the net torque is zero

D) Both the net force and the net torque equal zero

Answer 20

A) The net force is zero, but the net torque is not

There are 2 horizontal forces that point in opposite directions (so they cancel), and there are 2 vertical forces that point in opposite directions (so they cancel). Therefore, all forces cancel each other and the net force is zero.

All forces point in a counterclockwise direction, therefore net torque cannot be zero. In order for net torque to be zero, there must be a force in the clockwise direction that equals the force in the counterclockwise direction.

Question 21

As used in physics, the term equillibrium means zero __1___, whereas the term static equillibrium means zero __2___

Answer 21

1) acceleration (therefore velocity is constant)
2) velocity

Question 22

What’s the difference between translational equillibrium and rotational equillibrium?

Answer 22

translational equillibrium refers to when the forces cancel; therefore F_net = 0

rotational equillibrium refers to when the torques cancel; therefore T_net = 0

Question 23

What is inertia? What is rotational inertia?

Answer 23

inertia: resistance to acceleration (whether translational or rotational)

rotational inertia: an object’s resistance to rotational acceleration

Question 24

Equation for torque involving rotational inertia?

Answer 24

T = Ia

I = rotational inertia

a = rotational acceleration

Notice the similarity to F_net = ma. Torque is to rotational motion what Force is to translational motion. Similarly, rotational inertia, I, is to rotational motion what translational inertia (mass, m) is to translational motion.

Question 25

If object 1 has a greater rotational inertia than object 2, which object will be more difficult to rotate and therefore require a greater amount of torque to achieve the same rotational acceleration?

Answer 25

Object 1 will be more difficult to rotate and will require a greater amount of torque in order to achieve the same rotational acceleration

Question 26

Object 1 is heavier than object 2. If the same amount of torque is applied to both objects, which one will have greater rotational acceleration?

Answer 26

Object 2 will have greater rotational acceleration. When the same amount of torque is applied, as rotational inertia (mass) increases, rotational acceleration decreases. This can be seen from the equation T = Ia (T = torque, I = rotational inertia, a = angular acceleration)

Question 27

The farther the away the mass is from the axis of rotation, the __1__ the rotational inertia will be, which means the __2___ it will be to rotate this object upon its axis.

Answer 27

1) greater
2) more difficult

Question 28

If you have 2 balls of equal masses and equals sizes, but one ball holds most of its mass in the center, while the other ball holds most of its mass on the outer edges, which ball is considered to have greater rotational inertia? Which ball would be harder to rotate?

Answer 28

The ball with the weight distributed to its outer edges would have the greater rotational inertia, and therefore would be harder to move. If both balls were given the same push (aka were applied the same amount of torque), the ball with the weight distributed to its outer edges would have smaller rotational acceleration, and therefore would rotate more slowly.

Question 29

the center of mass is the same as the center of ____

Answer 29

gravity