a grind stone starts from rest and has constant angular accelaration 3rad/s2.(a)find the angular displacement?(b)the angular speed of the grind stone 20s later.?class11 state
b. V = a*t = 3rad/s^2 * 20s. = 60 rad/s.
To find the angular displacement of the grind stone, we can use the equation:
θ = ω0t + (1/2)αt^2
Where:
θ = angular displacement
ω0 = initial angular velocity (zero in this case)
α = angular acceleration
t = time
(a) Let's calculate the angular displacement:
Given:
α = 3 rad/s^2
ω0 = 0 rad/s
t = ? (unknown)
θ = ω0t + (1/2)αt^2
Since the initial angular velocity is zero (ω0 = 0), the equation simplifies to:
θ = (1/2)αt^2
Substituting the given value of α, we have:
θ = (1/2)(3)t^2
θ = (3/2)t^2
(b) To find the angular speed of the grind stone 20 seconds later, we can use the equation:
ω = ω0 + αt
Given:
α = 3 rad/s^2
ω0 = 0 rad/s
t = 20 s
ω = ω0 + αt
Substituting the given values, we have:
ω = 0 + 3(20)
ω = 60 rad/s
Therefore, the angular speed of the grind stone 20 seconds later is 60 rad/s.
To answer these questions, we need to use the equations of rotational motion.
(a) Finding the angular displacement:
The equation for angular displacement can be derived from the equation of rotational motion:
θ = ωi * t + (1/2) * α * t^2
Where:
θ = angular displacement (in radians),
ωi = initial angular velocity (in radians/second),
α = angular acceleration (in radians/second^2),
t = time (in seconds)
Given:
ωi = 0 (as the grind stone starts from rest),
α = 3 rad/s^2
Since the initial angular velocity is zero, the first term in the equation becomes zero.
θ = (1/2) * α * t^2
Substituting the given values, we have:
θ = (1/2) * 3 * t^2
θ = (3/2) * t^2
So, the angular displacement at any time t is given by the equation (3/2) * t^2.
(b) Finding the angular speed of the grind stone after 20 seconds:
The equation for angular velocity can also be derived from the equation of rotational motion:
ω = ωi + α * t
Given:
ωi = 0 (as the grind stone starts from rest),
α = 3 rad/s^2,
t = 20 s
Substituting the given values, we have:
ω = 0 + 3 * 20
ω = 60 rad/s
Therefore, the angular speed of the grind stone after 20 seconds is 60 radians/second.