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ddd [48]
3 years ago
14

A wheel with radius 36 cm is rotating at a rate of 19 rev/s.(a) What is the angular speed in radians per second? rad/s(b) In a t

ime interval of 5 s, what is the angle in radians through which the wheel rotates? rad(c) At t=10 s the angular speed begins to increase at a rate of 1.6 rad/s/s. At t=15 s, what is the angular speed in radians per second? rad/s(d) Through what angle in radians did the wheel rotate during the time between t=10 s and t=15 s? rad(e) If the wheel rolls along the ground without slipping, the instantaneous velocity of the atoms of the object that are momentarily in contact with the ground is zero. This zero-velocity condition implies that vCM=ω⁢R, where ω is the angular speed of the object, since the instantaneous speed of the contact point is vCM-ω⁢R. During the time between t=10 s and t=15 s, how far did the center of the wheel move, in meters? m
Physics
1 answer:
Sedaia [141]3 years ago
8 0

(a) 119.3 rad/s

The angular speed of the wheel is

\omega= 19 rev/s

we need to convert it into radiands per second. We know that

1 rev = 2 \pi rad

Therefore, we just need to multiply the angular speed of the wheel by this factor, to get the angular speed in rad/s:

\omega = 19 rev/s \cdot (2\pi rad/rev))=119.3 rad/s

(b) 596.5 rad

The angular displacement of the wheel in a time interval t is given by

\theta= \omega t

where

\omega=119.3 rad

and

t = 5 s is the time interval

Substituting numbers into the equation, we find

\theta=(119.3 rad/s)(5 s)=596.5 rad

(c) 127.3 rad/s

At t=10 s, the angular speed begins to increase with an angular acceleration of

\alpha = 1.6 rad/s^2

So the final angular speed will be given by

\omega_f = \omega_i + \alpha \Delta t

where

\omega_i = 119.3 rad/s is the initial angular speed

\alpha = 1.6 rad/s^2 is the angular acceleration

\Delta t = 15 s - 10 s = 5 s is the time interval

Solving the equation,

\omega_f = (119.3 rad/s) + (1.6 rad/s^2)(5 s)=127.3 rad/s

(d) 616.5 rad

The angle through which the wheel has rotated during this time interval is given by

\theta = \omega_i \Delta t + \frac{1}{2} \alpha (\Delta t)^2

Substituting the numbers into the equation, we find

\theta = (119.3 rad/s)(5 s) + \frac{1}{2} (1.6 rad/s^2) (5 s)^2=616.5 rad

(e) 222 m

The instantaneous speed of the center of the wheel is given by

v_{CM} = \omega R (1)

where

\omega is the average angular velocity of the wheel during the time t=10 s and t=15 s, and it is given by

\omega=\frac{\omega_i + \omega_f}{2}=\frac{127.3 rad/s+119.3 rad/s}{2}=123.3 rad/s

and

R = 36 cm = 0.36 m is the radius of the wheel

Substituting into (1),

v_{CM}=(123.3 rad/s)(0.36 m)=44.4 m/s

And so the displacement of the center of the wheel will be

d=v_{CM} t = (44.4 m/s)(5 s)=222 m

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