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 and . Here, the symbols are defined as follows:
 is the angular position of the particle at time .
 is the initial angular position of the particle.
 is the angular velocity of the particle at time .
 is the initial angular velocity of the particle.
 is the angular acceleration of the particle.
 is the time that has elapsed since the particle was located at its initial position.








Answer
General guidance
The equations of motion for rotational motion are, Here, is the final velocity, is the initial velocity, is the angular acceleration, is the time and is the angular displacement.
Stepbystep
Step 1 of 8
(A) The formula for is given to be, 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 terms, the quantity represented by is a function of time.
The quantity represented by is a function of time.
The formula, 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.
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 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.
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.
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, Since it is clearly seen that the formula for angular displacement at time contains terms, thus, the equation 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, Since it is clearly seen that the formula for angular velocity at time does not contain the term , hence, the equation is not an explicit function of time.
The equation is not an explicit function of time.
The formula for which is,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.
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 is, At time , angular velocity of particle B is, Angular acceleration of particle B is, The angular displacement of the particle B at will be, Substitute for , for and for in the above equation.
The equation, describes the angular position of particle B.
The formula was used to determine the equation describing the position of particle B.
Step 8 of 8
(H) The angular position of particle A is, Differentiate the above equation with respect to . …… (1) The angular position of particle B is, Differentiate the above equation with respect to . …… (2) Equate equations (1) and (2). Solve the equationfor t.
The time after which the angular velocity of particle A will be equal to the angular velocity of particle B is, .
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
The quantity represented by is a function of time.
The quantity represented by is not a function of time.
The quantity represented by is not a function of time.
The quantity represented by is a function of time.
The equation is not an explicit function of time.
The time variable represents the time elapsed from when the angular velocity equals until the angular velocity equals .
The equation, describes the angular position of particle B.
The time after which the angular velocity of particle A will be equal to the angular velocity of particle B is, .
Answer only
The quantity represented by is a function of time.
The quantity represented by is not a function of time.
The quantity represented by is not a function of time.
The quantity represented by is a function of time.
The equation is not an explicit function of time.
The time variable represents the time elapsed from when the angular velocity equals until the angular velocity equals .
The equation, describes the angular position of particle B.
The time after which the angular velocity of particle A will be equal to the angular velocity of particle B is, .