Question

1. A rotating top has a scratch 0.05 m from the vertical axis of rotation. In the time it takes the top to rotate 81 degrees, it rotates __ radians or __ revolutions. Furthermore, the scratch moves a length of arc of __in space. Give your answers to three decimal places.

2. A rotating top has a scratch 0.05 m from the vertical axis of rotation. It increases the angular velocity from 20 rad/s to 40 rad/s in 4 seconds. The angular acceleration of the top is __rad/s^2. The tangential acceleration of the scratch is __m/s^2. Give your answers to two decimal places.

3. A skater is spinning at 2 rev/s. she increases her angular velocity in 7 seconds until it is 8π rad/s. By what angle, in degrees, does she rotate in that time?

4. A skater is spinning at 0.6 rev/s. What angular acceleration is necessary. In units of rad/s^2, to increase her angular velocity to 9 rad/s while she rotates by an angle of 1000 degrees.

Answer 1

Question 1

Distance of the scratch from the vertical axis of rotation = R = 0.05 m

Angle through which the top rotates = = 81 degrees

Converting the angle to radians, (2 rad = 360 degrees)

= 1.414 rad

Converting the angle to revolutions, (360 degrees = 1 revolution)

= 0.225 rev

Length of the arc the the scratch moves = L

L = R (Here is in radians)

L = (0.05)(1.414)

L = 0.071 m

a) The top rotates through = 1.414 rad = 0.225 rev

b) The scratch moves through a length of arc of 0.071 m

Question 2

Distance of the scratch from the vertical axis of rotation = R = 0.05 m

Initial angular velocity of the scratch = ₁ = 20 rad/s

Angular velocity of the scratch after 4 sec = ₂ = 40 rad/s

Time period = T = 4 sec

Angular acceleration of the top =

₂ = ₁ + T

40 = 20 + (4)

= 5 rad/s²

Tangential acceleration of the scratch = a

a = R

a = (5)(0.05)

a = 0.25 m/s²

a) Angular acceleration of the top = 5 rad/s²

b) Tangential acceleration of the scratch = 0.25 m/s²

Question 3

Initial angular velocity of the skater = ₁ = 2 rev/s

Converting to rad/s, (1 rev = 2 rad)

₁ = 2 x (2) rad/s

₁ = 12.566 rad/s

Angular velocity of the skater after 7 sec = ₂ = 8 rad/s = 25.132 rad/s

Time period = T = 7 sec

Angular acceleration of the skater = (rad/s²)

₂ = ₁ + T

25.132 = 12.566 + (7)

= 1.795 rad/s²

Angle through which the skater rotates = (rad)

= ₁T + T²/2

= (12.566)(7) + (1.795)(7)²/2

= 131.94 rad

Converting to degrees, (2 rad = 360 degrees)

= 7560 degrees

a) Angle through which the skater rotates in that time = 7560 degrees

Question 4

Initial angular velocity of the skater = ₁ = 0.6 rev/s

Converting to rad/s, (1 rev = 2 rad)

₁ = 0.6 x (2) rad/s

₁ = 3.77 rad/s

Final angular velocity of the skater = ₂ = 9 rad/s

Angular acceleration of the skater = (rad/s²)

Angle through which the skater rotates = = 1000 degrees

Converting to radians, (2 rad = 360 degrees)

= 17.453 rad

₂² = ₁² + 2

(9)² = (3.77)² + 2(17.453)

= 1.91 rad/s²

Angular acceleration of the skater = 1.91 rad/s²