a bar on a hinge starts from rest and rotates with an angular acceleration α = 10 + 6t, where α is in rad/s and t is in seconds. determine the angle in radians through which the bar turns in the first 4.00s
Tisoy ka
To determine the angle in radians through which the bar turns in the first 4.00 seconds, we need to integrate the given angular acceleration expression with respect to time over the interval [0, 4.00].
The equation for angular acceleration is α = 10 + 6t.
To find the angular velocity (ω), we integrate the angular acceleration function:
ω = ∫(α) dt = ∫(10 + 6t) dt
ω = 10t + 3t^2/2 + C
To find the angle (θ), we integrate the angular velocity function:
θ = ∫(ω) dt = ∫(10t + 3t^2/2 + C) dt
θ = 5t^2 + t^3/2 + Ct + D
Since the bar starts from rest, the initial angular velocity and angle are 0:
ω(0) = 0 and θ(0) = 0
Substituting these initial conditions, we find:
0 = 0 + 0 + C --> C = 0
Therefore, the equation for the angle θ becomes:
θ = 5t^2 + t^3/2
Now we can find the angle through which the bar turns in the first 4.00 seconds:
θ(4.00) = 5(4.00)^2 + (4.00)^3/2
θ(4.00) = 5(16.00) + 8.00^2
θ(4.00) = 80.00 + 64.00
θ(4.00) = 144.00 radians
Therefore, the bar rotates through an angle of 144.00 radians in the first 4.00 seconds.
To determine the angle in radians through which the bar turns in the first 4.00 seconds, we need to integrate the angular acceleration function over the given time interval.
The equation for angular acceleration, α, is given as α = 10 + 6t, where α is in rad/s and t is in seconds.
To integrate α with respect to time, we first find the antiderivative of the function α = 10 + 6t:
∫ α dt = ∫ (10 + 6t) dt
Integrating each term separately, we get:
∫ α dt = ∫ 10 dt + ∫ (6t) dt
= 10t + 3t^2 + C
Here, C is the constant of integration.
Now, since we want to find the angle, we need to evaluate the above expression within the given time interval from t = 0 to t = 4.00 seconds:
Angle = [10t + 3t^2 + C] from 0 to 4.00
Substituting the upper and lower limits into the expression, we get:
Angle = [10(4.00) + 3(4.00)^2 + C] - [10(0) + 3(0)^2 + C]
= [40 + 3(16) + C] - [0 + 0 + C]
= 40 + 48 + C - C
= 88 radians
Therefore, the bar turns through an angle of 88 radians in the first 4.00 seconds.