12. Rotational Kinematics

Rotational Velocity & Acceleration

Rotational Motion

Angular Velocity and Acceleration

Angular Motion and Torque

Rotational Motion: An Explanation, Angular Displacement, Velocity and Acceleration

FIGURE EX4.24 shows the angular-position-versus-time graph for a particle moving in a circle. What is the particle’s angular velocity at (b) t = 4s<img src="https://lightcat-files.s3.amazonaws.com/problem_images/80e490a514c2915b-1677609889714.jpg" alt="" />

CALC The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct^3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = p/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s^2. (c) What are θ and the angular velocity when the angular acceleration is 3.50 rad/s^2?

Rotational velocity of Earth

Calculate the rotational velocity (in rad/s) of a clock’s minute hand.&nbsp;EXTRA:Calculate the rotational velocity (in rad/s) of a clock’s hour hand.

Anderson Video - Non-Uniform Circular Motion

 A person runs ¼ of the way around a circular lake in 5.6 minutes. What is the person’s angular speed? 

CALC A fan blade rotates with angular velocity given by ω_z(t) = g - bt^2, where g = 5.00 rad/s and b = 0.800 rad/s^3. (a) Calculate the angular acceleration as a function of time.

<math display="inline"><mrow><mn>3.2</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>, away from the center of the circle

<math display="inline"><mrow><mn>5.3</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>, toward the center of the circle

<math display="inline"><mrow><mn>5.3</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>, away from the center of the circle

<math display="inline"><mrow><mn>9.8</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>, down

 The car is not accelerating because it is travelling at a constant speed. 

<math display="inline"><mrow><mn>3.2</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>, toward the center of the circle

A car drives at a constant <math display="inline"><mrow><mn>15</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><mi mathvariant="normal" class="MathML-Unit">s</mi></mrow></mrow></math> along a curve on city street with a radius of <math display="inline"><mrow><mn>70</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">m</mi></mrow></math>. What is the acceleration of the car?

CALC The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct^3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = p/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s^2. (b) What is the angular acceleration when θ = p/4 rad?

The dwarf planet Ceres has a radius of <math display="inline"><mrow><mn>476</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">km</mi></mrow></math>, a mass of <math display="inline"><mrow><mn>9.5</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mn>20</mn></mrow></msup><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">kg</mi></mrow></math>, and a free fall acceleration due to gravity of <math display="inline"><mrow><mn>0.28</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mi mathvariant="normal" class="MathML-Unit">m</mi><mo>/</mo><msup><mi mathvariant="normal" class="MathML-Unit">s</mi><mn>2</mn></msup></mrow></mrow></math>. What speed will a rock have if it is orbiting Ceres, right on the surface? Ceres has no atmosphere – so there are no issues with air resistance – but assume that there are no mountains or large boulders to impede the orbit.

CALC A fan blade rotates with angular velocity given by ω_z(t) = g - bt^2, where g = 5.00 rad/s and b = 0.800 rad/s^3. (b) Calculate the instantaneous angular acceleration α_z at t = 3.00 s and the average angular acceleration α_av-z for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?

Rotational Motion: Kinematic Equations, Example Problems

CP CALC The angular velocity of a flywheel obeys the equation ω_z(t) = A + Bt2, where t is in seconds and A and B are constants having numerical values 2.75 (for A) and 1.50 (for B). (b) What is the angular acceleration of the wheel at (i) t = 0 and (ii) t = 5.00 s?

A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration.

CALC The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct^3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = p/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s^2. (a) Find a, b, and c, including their units.

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