Question

The 0.2 kg ball is blown through the smooth vertical circular tube whose shape is defined by r=(0.6sin?) m, where ? is in radians. If ?=(?t2) rad, where t is in seconds, determine the magnitude of force F exerted by the blower on the ball when t = 0.5 s.

Answer 1

Concepts and reason

According to Newton’s second law of motion the rate of change of momentum of the body must be equal to the net force applied to the body in the direction of motion. In simple words, the net force acting on a body must be equal to the mass of the body multiplied by the acceleration of the body.

Similarly, Newton’s second law for rotation states that the net torque acting on the body is equal to the mass moment of inertia of the body multiplied by the angular acceleration of the body.

When all the supports are removed by replacing them with forces that prevents the translation of body in a given direction that diagram is called free body diagram. When their resultant force and couples becomes equal to zero then the body is said to be in equilibrium.

To apply these equilibrium equations, we need to know the known and unknown forces that act on the body.

In this problem calculate the radial and angular acceleration of the ball and apply Newton’s second law of motion to calculate the value of unknown forces

Fundamentals

Suppose a body is loaded by some external force F and the mass of the body is m, the body is accelerating with a in the direction of force, then according to Newton’s second law:

Similarly, in case of rotation of body due to some external torque T, and the body is having angular acceleration of , then by Newton’s second law of motion:

The acceleration of a particle in cylindrical coordinate system can be written as:

Here, is the acceleration in radial direction, is the acceleration in azimuth direction or angular direction and is the acceleration in z direction.

Consider the angular displacement of the ball.

…… (1)

Substitute 0.5 s for t.

Differentiate equation (1) with respect time t.

…… (2)

Substitute 0.5 s for t.

Again, differentiate equation (2) with respect to time t.

Consider the radial displacement of the ball:

…… (3)

Substitute for .

Differentiate equation (3) with respect to .

…… (4)

Substitute for , for and for .

Again, differentiate equation (4) with respect to .

Substitute for , for and for .

Calculate the radial component of acceleration.

Substitute for , for r and for .

Calculate the angular component of acceleration.

Substitute for r, for , for and for .

Draw the free-body diagram of the ball at .

Apply Newton’s second law of motion in radial direction.

Substitute for , 0.2 kg for m, for g and for .

…… (5)

Apply Newton’s second law of motion in direction.

Substitute for , 0.2 kg for m, for g and for .

…… (6)

Add equation (5) and (6).

Ans:

The magnitude of force F exerted by the blower on the ball is .