Question

A block of 3-kg mass slides down a loop of 3-m radius with the coefficient of friction between

the block and loop being 0.25 at initial velocity (v) and enters a smooth horizontal plane, and
then compresses a spring of stiffness 0.25 kN/m as shown below. The spring is originally
unstretched. The normal acceleration is ignored when the block slides down the loop.
(a) Determine the minimum initial velocity of the block to ensure that the block can return
to its initial position (10 marks).
(b) Based on the above condition determine the compression of the spring when the block
touches the spring for the first time (10 marks)

Answer 1

To determine the minimum initial velocity of the block, use the net force and centripetal force. To determine the compression of the spring, use the conservation of mechanical energy.

Step-by-step explanation:

To determine the minimum initial velocity of the block to ensure that it can return to its initial position, we need to consider the forces acting on the block. When the block slides down the loop, the only force in the horizontal direction is the friction force. The friction force can be calculated as the product of the coefficient of friction and the normal force. Since the normal acceleration is ignored, the net force in the horizontal direction is equal to the friction force. The minimum initial velocity can be found by equating the net force to the centripetal force.

To determine the compression of the spring when the block touches it for the first time, we can use the conservation of mechanical energy. Initially, the block has kinetic energy, and this energy is converted into potential energy of the spring at maximum compression. The potential energy of the spring can be calculated using the formula ½ kx², where k is the spring stiffness and x is the compression of the spring. Equating the initial kinetic energy of the block to the potential energy of the spring will give us the compression of the spring.

Answer 2

(a) The minimum initial velocity of the block to ensure that the block can return to its initial position is 5.42 m/s.

(b) The compression of the spring is 0.6 m.

How to calculate the minimum initial velocity?

(a) The minimum initial velocity of the block to ensure that the block can return to its initial position is calculated by applying the following formula.

Potential energy = rotational KE + translational KE + work done by friction

mgr = ¹/₂Iω² + ¹/₂mv² + μmgr

where;

• I is the moment of inertia of the loop

mgr - μmgr = ¹/₂ (¹/₂mr²)(v/r)² + ¹/₂mv²

mgr - μmgr = ¹/₄mv² + ¹/₂mv²

mgr - μmgr = ³/₄mv²

gr - μgr = ³/₄v²

v² = (4 (gr - μgr)/3

v = √ (4 (gr - μgr)/3]

v = √ [(4 (9.8 x 3 - 0.25 x 9.8 x 3) / 3 ]

v = 5.42 m/s

(b) The compression of the spring is calculated as follows;

rotational KE + translational KE - work done by friction = elastic energy of the spring

³/₄mv² - μmgr = ¹/₂kx²

³/₄ x 3 x (5.42)² - (0.25 x 3 x 9.8 x 3) = ¹/₂ x (250) x (x²)

44.05 = 125x²

x² = 0.3524

x = √0.3524

x = 0.6 m