Question

f Question 1 (40 marks) (Compulsory) Answer each sub-question, a) to j). Long answers or explanations are not required. Full marks will be awarded for succinct, comect answers, whether as mathematical expressions, suitably labelled dlagrams, or brief text. If you use standard symbols in your answers there is no need to spend time defining them, unless asked to do so. a) Explain why the direction of the velocity of a particle is tangential to its trajectory. [4 marks] b) What is a Hodograph? Why is acceleration tangential to the Hodograph? [4 marks] c) For a rectangular coordinate system, we simply differentiate each component (in the x, y and z direction respectively) of the displacement vector with respect to time to obtain velocity. However, the unit vectors do not appear as differentiated with respect to time. Should they not be differentiated using the product rule, since the component in each direction is a vector equal to the magnitude of the component multiplied by the unit vector? Explain your answer [4 marks] d) For a rectangular coordinate system, show how a projectle can generate parabolic motion (only consider effect of constant gravity, ignore wind and other effects). [4 marks] e) Starting from the definition of angular velocity and acceleration, arrive at the relation do-α de where is the angular velocity, α is the angular acceleration and is the angular rotation. [4 marks] Consider the constrained motion of the system represented in Figure 1a. Derive the equation of motion of this system [4 marks g) Show, using an illustration, the direction of normal and tangential accelerations when a rigid body rotates about a foxed axis. [4 marks Figure 1 a h) What is the instantaneous centre of zero velocity? Show with an ustration. 4 marks) i) The two spheres, as shown in Figure 1b, each of mass m, are connected by the spring and hinged bars of negligible mass. The spheres are free to slide in the smooth guides up the incline of angle e Determine the acceleration ac of the centre Cof the spring [4 marks] Figure 1 b

Answer 1

a)

Velocity is time rate change of displacement. Rate of change is defined as derivative. We know that derivative of any curve gives its tangent at a point. So, velocity is tangential to a curve.

b)

Hodograph is proportional to velocity of the particle. Acceleration is defined as (time rate of change)derivative of velocity. So, acceleration is tangential to the velocity.

c)

In rectangular coordinate system, when we take a derivative of a vector, we take derivative of the components only. not of unit vector. This is because, motion in each of the direction is due to velocity in those directions only. So, the derivative of those unit vector directions is zero. But when rotational motion is involved this changes and so, cross products come in.

d)

Projectile motion:

$v_x = v cos\theta$

$v_y= v sin\theta$

$x= v_x*t = vcos\theta*t$

ucos

$y = v_y*t - \frac{1}{2}gt^2= vsin\theta*t - \frac{1}{2}gt^2$

-590 COS

$y = tan\theta*x - \frac{1}{2}g(\frac{x}{vcos\theta})^2$

this is an equation of a parabola symmetric about x axis.