Circular motion of a train magnitude and angle

[tyro008]

Homework Statement

A train is rounding a circular curve whose radius is 2.00 X 10$$_{}2$$ m. At one instant, the train has an angular acceleration of 1.50 X 10^-3 rad/s$$_{}2$$ and an angular speed of 0.0400 rad/s.
(a) Find the magnitude of the total acceleration (centripetal plus tangential) of the train.

(b) Determine the angle of the total acceleration relative to the radial direction.

Homework Equations

a$$_{}NET$$ = $$\sqrt{}ac2^{}_{} + at2^{}$$
and Ac is the centripetal acceleration, At is the tangential acceleration

The Attempt at a Solution

i tried using this equation, but i couldn't find At. to find this, you need delta velocity divided by delta time, but there is no time and i don't know if angular speed is the same thing as regular velocity

 

[grassblade]

You can find At by using the formula: angular acceleration = tangential acceleration / radius.

I'm not sure how to find the angle though...

 

[thomate1]

.

I would like to clarify that angular speed is not the same as regular velocity. Angular speed is a measure of how fast an object is rotating, while velocity is a measure of how fast an object is moving in a straight line. In this case, the angular speed of the train tells us how fast it is rotating around the circular curve.

To find the tangential acceleration, we can use the equation At = r x alpha, where r is the radius and alpha is the angular acceleration. Plugging in the values given, we get At = (2.00 X 10^-2 m) x (1.50 X 10^-3 rad/s^2) = 3.00 X 10^-5 m/s^2. This tells us that the train is also accelerating tangentially as it goes around the curve.

To find the magnitude of the total acceleration, we can use the Pythagorean theorem: a_net = √(Ac^2 + At^2). Plugging in the values, we get a_net = √((0.0400 rad/s)^2 + (3.00 X 10^-5 m/s^2)^2) = 0.0400 rad/s. This is the total acceleration of the train at that instant.

To determine the angle of the total acceleration relative to the radial direction, we can use the inverse tangent function: θ = tan^-1(At/Ac). Plugging in the values, we get θ = tan^-1((3.00 X 10^-5 m/s^2)/(0.0400 rad/s)) = 0.00238 radians or 0.14 degrees. This tells us that the total acceleration is at an angle of 0.14 degrees relative to the radial direction.