9709/Oct Nov/43/2011/Q6

The diagram shows a ring of mass 2 kg threaded on a fixed rough vertical rod. A light string is attached to the ring and is pulled upwards at an angle of 30◦ to the horizontal. The tension in the string is T N. The coefficient of friction between the ring and the rod is 0.24. Find the two values of T for which the ring is in limiting equilibrium. 

Solution:

Reference: PYQ - Oct/Nov 2011 Paper 43 Q6

9709/Oct Nov/43/2011/Q4

ABC is a vertical cross-section of a surface. The part of the surface containing AB is smooth and A is 4m higher than B. The part of the surface containing BC is horizontal and the distance BC is 5m (see diagram). A particle of mass 0.8 kg is released from rest at A and slides along ABC. Find the speed of the particle at C in each of the following cases.

(i) The horizontal part of the surface is smooth.

(ii) The coefficient of friction between the particle and the horizontal part of the surface is 0.3.

Solution:

(i) 

mgh = 0.8 ×  10 × 4   = 32

For using ½ mv^2 = PE

[½ 0.8v^2 = 32]

Speed at C, v = 8.94 ms^–1

(ii) 

R = mg

= 0.8 x 10

8 N

Friction force, 

F = μ mg

   = 0.3 (8)

   = 2.4 N

Force from B to C = 2.4N (opposite direction of friction force)

Work done from B to C 

= 2.4 x 5

= 12 J

Conservation of Energy

½ mv^2 + 12 = PE

½ mv^2 = 32 - 12

Speed at C, v = 7.07 ms^-1

Reference: PYQ - Oct/Nov 2011 Paper 43 Q4

9709/May June/2011/42/Q6

A small smooth ring R, of mass 0.6 kg, is threaded on a light inextensible string of length 100 cm. One end of the string is attached to a fixed point A. A small bead B of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through A. The system is in equilibrium with B at a distance of 80 cm from A (see diagram).

(i) Find the tension in the string. 

(ii) Find the frictional and normal components of the contact force acting on B. 

(iii) Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q6

9709/May June/2011/42/Q3

The velocity-time graph shown models the motion of a parachutist falling vertically. There are four
stages in the motion:
• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.

(i) Show that the total distance fallen is 1048m.
The weight of the parachutist is 850N.
(ii) Find the upward force on the parachutist due to the parachute, during the second stage.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q3

9709/May June/2011/42/Q2

An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is 3ms−1 and its speed at the bottom of the plane is 8ms−1. The work done against the resistance to motion of the object is 120 J. Find the height of the top of the plane above the level of the bottom.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q2

9709/Oct Nov/2002/4/Q4

Two particles A and B are projected vertically upwards from horizontal ground at the same instant.

The speeds of projection of A and B are 5ms^-1 and 8ms^-1 respectively. Find

(i) the difference in the heights of A and B when A is at its maximum height,

(ii) the height of A above the ground when B is 0.9m above A.

Solution:

Reference: PYQ - Oct/Nov 2002 Paper 4 Q4

9709/Oct Nov/2002/4/Q3

A light inextensible string has its ends attached to two fixed points A and B, with A vertically above B. A smooth ring R, of mass 0.8 kg, is threaded on the string and is pulled by a horizontal force of magnitude X newtons. The sections AR and BR of the string make angles of 50^o and 20^o  respectively with the horizontal, as shown in the diagram. The ring rests in equilibrium with the string taut. Find

(i) the tension in the string,

(ii) the value of X.

Solution:

Reference: PYQ - Oct/Nov 2002 Paper 4 Q3

Mechanics 1 Modelling with Friction Question 11

A chute at a water sports centre has been designed so that swimmers first slide down a steep part which is 10 m long and at an angle of 40° to the horizontal. They then come to a 20 m section with a gentler slope, 11° to the horizontal, where they travel at constant speed.

(i) Find the coefficient of friction between a swimmer and the chute.

(ii) Find the acceleration of a swimmer on the steep part.

(iii) Find the speed at the end of the chute of a swimmer who starts at rest.

(You may assume that no speed is lost at the point where the slope changes.)

An alternative design of chute has the same starting and finishing points but has a constant gradient.

(iv) With what speed do swimmers arrive at the end of this chute?

Solution:

Mechanics 1 Modelling with Friction Question 9

In each of the following situations a brick is about to slide down a rough inclined plane. Find the unknown quantity.
(i) The plane is inclined at 30° to the horizontal and the brick has mass 2 kg: find μ.
(ii) The brick has mass 4 kg and the coefficient of friction is 0.7: find the angle of the slope.
(iii) The plane is at 65° to the horizontal and the brick has mass 5 kg: find μ.
(iv) The brick has mass 6 kg and μ is 1.2: find the angle of slope.

Solution: