Test: Curvilinear Motion - Civil Engineering (CE) MCQ

When a body moves along a curved path a net force acts on the body that is directed towards the center of curvature. This force is called the centripetal Force, F_C. 
F_C = mV²/r
where, m is body mass, V is magnitude of the velocity, r is radius of curvature.

Calculation:
Given:
Case 1:
The car A is running on a plane bridge as shown -

The normal force F_A = mg 
Case 2:-
The car B is running on a convex bridge as shown - 

Now this is a case of plane circular motion and in this case, the net centripetal force acts towards the center of curvature.
So the normal force

Case 3:
The car C is running on a concave bridge as shown - 

In this case, there is also a net centripetal force acting towards the center of curvature.
So the normal force F_C

Hence, by looking at all three equations,
F_C > F_A > F_B 
we can say that the contact force F_C is the maximum among all three forces.

Velocity:
There is the only a tangential component of velocity in the curvilinear motion.

Let the unit vector in the tangential direction be denoted as 

Here, v is nothing but the derivative of a position vector

Here, the first term represents the tangential component of acceleration and the second term represents the radial component of acceleration

So, the radial component of acceleration = r˙θ˙
And the velocity component in a radial direction is 0.

Angular acceleration is the rate of change of angular velocity

Calculation:
Given

Radial and Transverse co-ordinates:

• In this system, the position of the particle is defined by the polar coordinates r and θ
• Further, the velocity and acceleration of the particle are resolved into components along and perpendicular to the position vector  These components are called:
  • Radial components denoted by
  • Transverse components denoted by 

The unit vectors of  are represented by  respectively. Here A₁A₂ and B₁B₂ are perpendicular to the x-axis and are drawn from the endpoints A₁ and B₁ of the unit vectors  respectivel
Furthur: 
That gives: A₁A₂ = Sin θ and B₁B₂ = Cos θ, PA₂ = Cos θ and PB₂ = Sin θ
Applying the triangle law of addition of vectors

Velocity Component: When the above identities are differentiated with respect to time t.


In terms of the position vector , the velocity vector is defined as

Acceleration:

Substituting the values for  in acceleration.

The radial component of acceleration,

Centripetal Acceleration (a_c): 

• Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.
• It always acts on the object along the radius towards the center of the circular path.
• The magnitude of centripetal acceleration,

a = v²/r
Where v = velocity of the object and r = radius
Tangential acceleration (a_t):

• It acts along the tangent to the circular path in the plane of the circular path.
• Mathematically Tangential acceleration is written as

Where α = angular acceleration and r = radius
CALCULATION:
Given – v = 10 m/s, r = 25 m and a_t = 3 m/s²

• Net acceleration is the resultant acceleration of centripetal acceleration and tangential acceleration i.e.,

Centripetal Acceleration (a_c):

Hence, net acceleration 

Tangential acceleration = r × angular acceleration

Now,

Now,
Normal acceleration 

∴ Resultant acceleration 

Distance covered by translation, S = u × t + (1/2)at²
u is the initial motion, t is the time and a is the acceleration
Calculation:
Given u = 0, a = 5 m/s² and t = 5 s