a truck travels along the horizontal circular curve of radius r=60m with a constant speed v=20m/s. Determine the angular rate of rotation theta of the radial line r and the magnitude of the truck's acceleration.
To determine the angular rate of rotation, also known as angular velocity (θ), we can use the relationship between linear velocity (v) and angular velocity (θ) in circular motion.
The linear velocity is given as v = 20 m/s. We can use the formula:
v = r * θ,
where r is the radius of the circular curve.
Rearranging the formula to solve for θ, we have:
θ = v / r.
Substituting the given values, θ = 20 m/s / 60 m = 1/3 rad/s.
So, the angular rate of rotation (θ) of the radial line r is 1/3 rad/s.
Next, to find the magnitude of the truck's acceleration, we can use the centripetal acceleration formula:
a = v^2 / r.
Plugging in the given values, we have:
a = (20 m/s)^2 / 60 m = 400 m^2/s^2 / 60 m = 20/3 m/s^2.
Therefore, the magnitude of the truck's acceleration is 20/3 m/s^2.
The truck's acceeration is entirely centripetal, and the magnitude is
a = v^2/r
The angular rotation rate, in radians per second, is
w = v/r