A particle moves in a plane with trajectory given by the polar equation,
r = a(1 + cos �);where a is a positive constant Suppose that the particle moves such that �theta(t) =omega for all t; that is, the angular velocity �theta is constant.Show that the
acceleration has radial component -aomega^2 (2 cos �+1)and transverse component
-2aomega^2sintheta
first find velocity and acceleration then automatically answer will come out
To find the acceleration of the particle, we need to determine its radial and transverse components. Let's start by calculating the velocity of the particle.
The velocity of a particle in polar coordinates can be obtained by differentiating the position vector with respect to time. In this case, the position vector r is given by:
r = a(1 + cosθ)
To differentiate with respect to time, we need to express r in terms of t. Since θ is constant, we can express it as θ = ωt, where ω is the angular velocity.
Substituting θ = ωt into the equation for r, we get:
r = a(1 + cos(ωt))
Now, we can differentiate r with respect to t to find the velocity:
v = dr/dt = aω(-sin(ωt))
Notice that we have used the chain rule of differentiation and the derivative of cos(ωt) is -ωsin(ωt).
Next, let's find the acceleration of the particle by differentiating the velocity vector with respect to time:
a = dv/dt = d(aω(-sin(ωt)))/dt
Using the chain rule again, we obtain:
a = -aω^2(cos(ωt))
Now, we need to express the acceleration in terms of θ instead of t. Since θ = ωt, we can rewrite the expression for the acceleration as:
a = -aω^2(cosθ)
The radial component of the acceleration is the component in the radial direction, which is given by the negative of the acceleration magnitude multiplied by cosθ:
Radial component = -aω^2(cosθ)
The transverse component of the acceleration is the component perpendicular to the radial direction, which is given by the negative of the acceleration magnitude multiplied by sinθ:
Transverse component = -aω^2(sinθ)
Therefore, from the expressions above, we can conclude that the acceleration has a radial component of -aω^2(2cosθ + 1) and a transverse component of -2aω^2sinθ.