A train is travelling along the circular curve of radius 200m at the instant shown its angular rate of rotation is 10 rad/s which is decresing at 0.2 rad/s determine the magnitude of the trains velocity & acceleration at this instsnt...
To determine the magnitude of the train's velocity and acceleration at the given instant, we'll need to use some basic principles of circular motion.
1. Velocity of the train:
The velocity of an object moving in a circle is given by the formula: v = rω, where v is the velocity, r is the radius of the circle, and ω is the angular velocity. In this case, the radius is given as 200m, and the angular velocity is given as 10 rad/s. Plugging in these values, we get:
v = 200m * 10 rad/s = 2000 m/s.
So, at this instant, the magnitude of the train's velocity is 2000 m/s.
2. Acceleration of the train:
The acceleration of an object moving in a circle is given by the formula: a = rω^2, where a is the acceleration, r is the radius of the circle, and ω is the angular velocity. However, since the angular velocity is decreasing at a rate of 0.2 rad/s, we need to consider the instantaneous angular velocity to find the acceleration.
To find the instantaneous angular velocity, we subtract the rate of decrease (0.2 rad/s) from the given angular velocity (10 rad/s):
Instantaneous angular velocity = 10 rad/s - 0.2 rad/s = 9.8 rad/s.
Plugging this value into the formula for acceleration, we get:
a = 200m * (9.8 rad/s)^2 = 38040 m/s^2.
So, at this instant, the magnitude of the train's acceleration is 38040 m/s^2.