The drill bit of a variable speed electric drill has a constant angular acceleration of 2.50 rad/s^2. the initial angular speed of the bit is 5.00 rad/s. after 4.00s, (a) what angle has the bit turned through and (b) what is the bit's angular speed?
To solve this problem, we can use the equations of angular motion:
(a) The formula to calculate the angle turned by an object is given by:
θ = ω0t + (1/2)αt^2
where:
θ = angle turned
ω0 = initial angular speed
α = angular acceleration
t = time
Plugging in the values given, we have:
θ = (5.00 rad/s)(4.00 s) + (1/2)(2.50 rad/s^2)(4.00 s)^2
Simplifying this equation, we get:
θ = 20.00 rad + 10.00 rad
Therefore, the angle turned by the bit is:
θ = 30.00 rad
(b) The formula to calculate the final angular speed is given by:
ω = ω0 + αt
Plugging in the values given, we have:
ω = 5.00 rad/s + (2.50 rad/s^2)(4.00 s)
Simplifying this equation, we get:
ω = 5.00 rad/s + 10.00 rad/s
Therefore, the bit's angular speed is:
ω = 15.00 rad/s
To find the answer to these questions, we can use the equations of rotational motion. Let's start with part (a), finding the angle the bit has turned through.
The equation that relates angular displacement (θ) with initial angular velocity (ω₀), angular acceleration (α), and time (t) is:
θ = ω₀t + (1/2)αt²
In this case, we have:
ω₀ = 5.00 rad/s (initial angular speed)
α = 2.50 rad/s² (angular acceleration)
t = 4.00 s (time)
Using these values in the equation, we can calculate θ:
θ = (5.00 rad/s)(4.00 s) + (1/2)(2.50 rad/s²)(4.00 s)²
Calculating this expression gives us:
θ = 20.00 rad + 20.00 rad
θ = 40.00 rad
Therefore, the bit has turned through an angle of 40.00 radians after 4.00 seconds.
Moving on to part (b), finding the bit's angular speed after 4.00 seconds.
The equation that relates final angular velocity (ω) with initial angular velocity (ω₀), angular acceleration (α), and time (t) is:
ω = ω₀ + αt
Plugging in the given values, we have:
ω = 5.00 rad/s + (2.50 rad/s²)(4.00 s)
Calculating this expression gives us:
ω = 5.00 rad/s + 10.00 rad/s
ω = 15.00 rad/s
Therefore, the bit's angular speed after 4.00 seconds is 15.00 rad/s.
(a) just as with linear motion, 5t + 1/2 at^2
(b) since the speed increases by 2.5rad/s every second, it will have increased by 10 rad/s after 4 seconds.
a. 15rad/s * 4s = rad.
b. V = Vo + a*t = 5 + 2.5*4 = rad/s.