Question

The device at right is used to quickly lift the block shown. The free rope is pulled downward af a constant speed of 2 m/s for 15 s. The inner and outer diameters of the big pulley are 0.40 m and 0.90 m. and the inner and outer diameters of the small pulley are 0.2 m and 0.6 m Motion Information Big Pulley Event 1: Event 2: Little Pulley Event 1: Event 2: Block Event 1: Event 2: - 5 0= az - Mathematical Analysis

Solve for all unknown quantities

Answer 1

IBO WB q=2 m/s

Velocity with which the string is pulled: for 15 s

Inner and outer diameter of the big pulley 0.4 m 0.9 m

Inner and outer diameter of the small pulley 0.2 m 0.6 m

Outer radius of the big pulley

Inner radius of the big pulley

Outer radius of the small pulley

Inner radius of the small pulley

Since there is no acceleration on the string, the tension on the string will balance the weight of the block and there are no unbalanced forces. $\alpha_1=\alpha_2=a_1=0$

Angular velocity of the bigger pulley

[$v=r\omega$]

Since the velocity is on the inner radius, $\omega_1=\omega_B=\frac{v_a}{r_{BI}}=\frac{2}{0.2}=10\ rad/s$

The tangential velocity acting of the outer radius of the big pulley,

$v_{BO}=\omega_{B}\times r_{BO}=10 \times 0.45=4.5\ m/s$ [Note- the angular velocity of a rotating object is constant everywhere on it, whereas the tangential velocity at a point depends on the distance of the point from the centre]

The velocity of the part of the string between two pulleys is the same as the velocity of the outer rim of the bigger pulley $v_{SI}=v_{BO}$

The angular velocity of the smaller pulley,

$\omega_2=\omega_S=\frac{v_{SI}}{r_{SI}}=\frac{4.5}{0.1}=45\ rad/s$

The velocityof the outer rim of the small pulley,

$v_1=v_{SO}=\omega_S\times r_{SO}=45\times 0.3=13.5\ m/s$

The equations of motion:

$\theta_2=\theta_1+\omega_{B}t$ [Big pulley]

$\theta_2=\theta_1+\omega_{S}t$ [Small pulley]

$r_2=r_1+v_{SO}t$ [Block]

Since there are no accelerations,

$\omega_2=\omega_1=10\ rad/s$ [Big pulley]

$\omega_2=\omega_1=45\ rad/s$ [Small pulley]

[Block]

The info:

Big Pulley		Small Pulley		Block
Event 1	Event 2	Event 1	Event 2	Event 1	Event 2
$t_1=0$		$t_1=0$		$t_1=0$
$\omega_1=10\ rad/s$	$\omega_2=10\ rad/s$	$\omega_1=45\ rad/s$	$\omega_2=45\ rad/s$	$\v_1=13.5\ m/s$
$\theta_1=0$	$\theta_2=150\ rad$	$\theta_1=0$	$\theta_2=675\ rad$	$r_1=0$
$\alpha_1=0$	$\alpha_2=0$	$\alpha_1=0$	$\alpha_2=0$	$a_1=0$	$a_2=0$