A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a small block of mass 4 kg, which hangs freely (see diagram). The system is released from rest. During the subsequent motion, the block experiences a constant resistance to its motion, of magnitude RN. Given that the angular speed of the disc after it has turned through 2 radians is 5 rad s⁻¹, find R and the tension in the string.

Photo AI

A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre - CIE - A-Level Further Maths - Question 3 - 2013 - Paper 1

Question 3

A-uniform-disc,-of-mass-2-kg-and-radius-0.2-m,-is-free-to-rotate-in-a-vertical-plane-about-a-smooth-horizontal-axis-through-its-centre-CIE-A-Level Further Maths-Question 3-2013-Paper 1.png

A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensib... show full transcript

Worked Solution & Example Answer:A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre - CIE - A-Level Further Maths - Question 3 - 2013 - Paper 1

Step 1

Find the equation of motion for disc:

96%

114 rated

Answer

Let's start with applying the conservation of energy to the system. The gravitational potential energy lost by the block is converted into the kinetic energy of both the block and the disc, and work done against the resistance. Given that the potential energy lost is given by the formula:

$PE = mgh$

We have the block of mass 4 kg falling a height given by the arc through which the disc rotates:

$h = r(1 - ext{cos}( heta))$

where $r$ is the radius of the disc and $heta$ the angle in radians. After falling 2 radians, we can compute this height:

$h = 0.2(1 - ext{cos}(2))$

Step 2

Substitute to find T:

99%

104 rated

Answer

Using the equation of motion for the disc:

$\tau = I\alpha$

where $au$ is the torque, $I$ is the moment of inertia of the disc and $eta$ is the angular acceleration. The moment of inertia for a disc is given by:

$I = \frac{1}{2}mr^2$

Substituting the values gives:

$I = \frac{1}{2}(2)(0.2)^2 = 0.04 \text{ kg m}^2$

The angular acceleration can be determined from the change in angular speed:

$\alpha = \frac{\Delta \omega}{\Delta t}$

Step 3

Find equation of motion for block:

96%

101 rated

Answer

The net force acting on the block can be expressed as:

$F_{net} = mg - T - R$

where $F_{net} = ma$ , and we can replace $m = 4\text{kg}$ and the acceleration $a = \frac{g - T}{4}$ . Substituting known values will allow us to find T and R.

Step 4

Substitute to find R:

98%

120 rated

Answer

Using the relation of angular velocity to linear speed:

$v = r\omega$

and substituting for $heta$ gives:

$R = \frac{(4g - T)}{4}$

Using g = 9.81 m/s², we can evaluate for R.

Step 5

Find tension in the string:

97%

117 rated

Answer

Using the previously established relationship and resolving the equation will lead us to find T. Rearranging our equations from earlier will allow us to express T in terms of R calculated earlier and then substituting those values will yield the final results.

A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre - CIE - A-Level Further Maths - Question 3 - 2013 - Paper 1

Worked Solution & Example Answer:A uniform disc, of mass 2 kg and radius 0.2 m, is free to rotate in a vertical plane about a smooth horizontal axis through its centre - CIE - A-Level Further Maths - Question 3 - 2013 - Paper 1

Find the equation of motion for disc:

Substitute to find T:

Find equation of motion for block:

Substitute to find R:

Find tension in the string:

Join the A-Level students using SimpleStudy...

Other A-Level Further Maths topics to explore