True or false: the quantity represented by θ is a function of time (i.e. is not constant).

Rotational motion with a constant nonzero acceleration is not uncommon in the world around us. For instance, many machines have spinning parts. When the machine is turned on or off, the spinning parts tend to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as theta (t) = theta_0 +omega_0t + frac{1}{2}alpha t^2 and omega (t) = omega_0 + alpha t. Here, the symbols are defined as follows:

  • theta(t) is the angular position of the particle at time t.
  • theta_0 is the initial angular position of the particle.
  • omega(t) is the angular velocity of the particle at time t.
  • omega_0 is the initial angular velocity of the particle.
  • alpha is the angular acceleration of the particle.
  • t is the time that has elapsed since the particle was located at its initial position.
In answering the following questions, assume that the angular acceleration is constant and nonzero: alpha neq 0.
A)
True or false: The quantity represented by theta is a function of time (i.e., is not constant).
true
false
B)
True or false: The quantity represented by theta_0 is a function of time (i.e., is not constant).
true
false
C)
True or false: The quantity represented by omega_0 is a function of time (i.e., is not constant).
true
false
D)
True or false: The quantity represented by omega is a function of time (i.e., is not constant).
true
false
E)
Which of the following equations is not an explicit function of time t? Keep in mind that an equation that is an explicit function of time involves t as a variable.
theta=theta_0+omega_0 t+frac{1}{2}alpha t^2
omega=omega_0+alpha t
omega^2=omega_0^2+2alpha(theta-theta_0)
F)
In the equation omega=omega_0+alpha t, what does the time variable t represent?
Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.
the moment in time at which the angular velocity equals omega_0
the moment in time at which the angular velocity equals omega
the time elapsed from when the angular velocity equals omega_0 until the angular velocity equals omega
G)
Consider two particles A and B. The angular position of particle A, with constant angular acceleration, depends on time according to theta_{rm A}(t)=theta_0+omega_0t+frac{_1}{^2}alpha t^2. At time t=t_1, particle B, which also undergoes constant angular acceleration, has twice the angular acceleration, half the angular velocity, and the same angular position that particle A had at time t=0.
Which of the following equations describes the angular position of particle B?
theta_{rm B}(t)=theta_0+2omega_0t+frac{1}{4}alpha t^2
theta_{rm B}(t)=theta_0+frac{1}{2}omega_0t+alpha t^2
theta_{rm B}(t)=theta_0+2omega_0(t-t_1) +frac{1}{4}alpha (t-t_1)^2
theta_{rm B}(t)=theta_0+frac{1}{2}omega_0(t-t_1)+alpha (t-t_1)^2
theta_{rm B}(t)=theta_0+2omega_0(t+t_1) +frac{1}{4}alpha (t+t_1)^2
theta_{rm B}(t)=theta_0+frac{1}{2}omega_0(t+t_1)+alpha (t+t_1)^2
H)
How long after the time t_1 does the angular velocity of particle B equal that of particle A?
frac{omega_0}{4alpha}
frac{omega_0+4alpha t_1}{2alpha}
frac{omega_0+2alpha t_1}{2alpha}
The two particles never have the same angular velocity.

Answer

General guidance

Concepts and reason
The concept required to solve the given problem is rotational equations of motion. Use the rotational equations of motion together with the definition of angular velocity and displacement to derive the various conclusions.

Fundamentals

The equations of motion for rotational motion are, o² = ² +2ad 0=0, +at 0=0,1 +5 at Here, is the final velocity, is the initial velocity, is the angular acceleration, is the time and is the angular displacement.

Step-by-step

Step 1 of 8

(A) The formula for is given to be, အို+in+2=(1) Here, is the initial displacement, is the time, is the initial velocity and is the angular acceleration. Since it is clearly seen that the formula for angular displacement at time contains 1 and 2 terms, the quantity represented by is a function of time.

Part A

The quantity represented by is a function of time.


The formula, အို+in+2=(1)was used to determine whether the quantity depends on time or not.

Step 2 of 8

(B) The quantity is the initial angular position of the particle which is a constant and hence will not change with time. Thus, the quantity is independent on time.

Part B

The quantity represented by is not a function of time.


The formula, အို+in+2=(1)was used to determine whether the quantity depends on time or not.

Step 3 of 8

(C) The quantity is the initial angular velocity of the particle which is a constant and hence will not change with time. Thus, the quantity is independent on time.

Part C

The quantity represented by is not a function of time.


The formula, was used to determine whether the quantity depends on time or not.

Step 4 of 8

(D) The formula for is given to be, Here, is the time, is the initial velocity and is the angular acceleration. Since it is clearly seen that the formula for angular velocity at time contains the term . Thus, quantity represented by is a function of time.

Part D

The quantity represented by is a function of time.


The formula, was used to determine whether the quantity depends on time or not.

Step 5 of 8

(E) The formula for is given as, အို+in+2=(1) Since it is clearly seen that the formula for angular displacement at time contains 1 and 2 terms, thus, the equation အို+in+2=(1)is a function of time. The formula for is given to be, Since it is clearly seen that the formula for angular velocity at time contains the term , thus, the equation is a function of time. The formula for is also given to be, m’ = ,* + 2a(0-0) Since it is clearly seen that the formula for angular velocity at time does not contain the term , hence, the equation m’ = ,* + 2a(0-0)is not an explicit function of time.

Part E

The equation m’ = ,* + 2a(0-0)is not an explicit function of time.


The formula for which is,m’ = ,* + 2a(0-0)was used to determine whether equation is an explicit function of time or not.

Step 6 of 8

(F) The formula for is given to be, Here, the variable represents the time elapsed from when the angular velocity equals until the angular velocity equals . It is the time taken by the particle to reach its final angular velocity starting from initial angular velocity.

Part F

The time variable represents the time elapsed from when the angular velocity equals until the angular velocity equals .


The definition of time together with the formula, was used to derive the correct conclusion out of the given statements.

Step 7 of 8

(G) The angular displacement of the particle A at t=0 s is, 0.(t)= 0,+ @xt+-ar At time , angular velocity of particle B is, m
= Angular acceleration of particle B is, ag = 2a The angular displacement of the particle B at will be, + 2 + 0 = (1) * Substitute for , for and for in the above equation. 6, (༦)=6, +(༔ )(t=༣)+༩(༦=༣)

Part G

The equation, 6, (༦)=6, +(༔ )(t=༣)+༩(༦=༣)describes the angular position of particle B.


The formulaအို+in+2=(1) was used to determine the equation describing the position of particle B.

Step 8 of 8

(H) The angular position of particle A is, 0.(t)= 0,+ @xt+-ar Differentiate the above equation with respect to . 0.(t)=0, +at …… (1) The angular position of particle B is, 6, (༦)=6, +(༔ )(t=༣)+༩(༦=༣) Differentiate the above equation with respect to . ex (1)=(༧༠)+2a(༦=༩) …… (2) Equate equations (1) and (2). C+24(11) = 0, +at Solve the equationC+24(11) = 0, +atfor t.

1= 0, +2αι,
2α

Part H

The time after which the angular velocity of particle A will be equal to the angular velocity of particle B is, 1= 0, +2αι,
2α.


The formula, was used to determine the time at which the angular velocity of particle A will be equal to that of the angular velocity of particle B.

Answer

Part A

The quantity represented by is a function of time.

Part B

The quantity represented by is not a function of time.

Part C

The quantity represented by is not a function of time.

Part D

The quantity represented by is a function of time.

Part E

The equation m’ = ,* + 2a(0-0)is not an explicit function of time.

Part F

The time variable represents the time elapsed from when the angular velocity equals until the angular velocity equals .

Part G

The equation, 6, (༦)=6, +(༔ )(t=༣)+༩(༦=༣)describes the angular position of particle B.

Part H

The time after which the angular velocity of particle A will be equal to the angular velocity of particle B is, 1= 0, +2αι,
2α.


Leave a Comment