Posted: December 21st, 2015

# If the ladder is just on the point of slipping at both the upper and lower ends, what can you say about the pair of forces acting at each of the top and bottom of the ladder?

Please do not repeat the question text from this document in your eTMA submission as this is likely to be registered as possible plagiarism by our automatic detection system, and may lead to delays in returning your assignment.

In answering these questions take care to express all final answers to an appropriate number

of significant figures, but also ensure that any intermediate values, or answers carried forward

to another part, have sufficient significant figures to avoid propagation of rounding errors.

Take care to show your working throughout, and always indicate any principles or definitions

you are using. We recommend that you look over Section 3 of the Introduction and Guide

before starting work on this assignment.

This assignment covers: Unit 5, 6, and 7.

This TMA is designed to test your achievement of the following learning outcomes for this module:

Question 1

This question is worth 10 marks and is designed to test your understanding of work, energy

and power (Unit 5).

A stone used in the sport of curling has a mass of 18.0 kg and is initially at rest, sitting on a

ﬂat ice surface. It is pushed with a constant force of magnitude 24.0N over a distance of

4.00 m. (You may assume that any friction between the stone and the ice is negligible.)

(a)

(i) What is the acceleration of the stone and what is the work done in accelerating it?

(ii) By considering the conservation of energy of the stone, calculate the final speed of the

stone.

(b) How much power is supplied by the person pushing the stone at the beginning of the

push and how much is supplied at the end of the push? (2 marks)

(c) Calculate the duration of the push from the acceleration and final speed. Compare this

with the time calculated from the average power supplied during the push and the amount of

work done.

Question 2

This question is worth 16 marks and is designed to test your understanding of the

conservation of energy and momentum (Unit 6).

Consider two railway trucks that can travel along a straight railway track which defines the

x-axis. A laden truck of mass M =6.6 × 10 kg and an empty truck of unknown mass m

approach each other at constant velocity. The laden truck has an initial velocity component

of u x =+2.0ms

which is half that of the initial velocity component of the empty truck (but in the opposite direction). After an elastic collision, the velocity component of the laden truck is halved, but it continues in the same direction. The unknown post-collision velocity component of the empty truck is v x

(a) Begin by clearly setting out the various masses and velocity components involved, then

write down equations for the conservation of linear momentum and the conservation of kinetic energy, in terms of the variables M,m, u x

and v only.

(b) Rearrange the momentum equation into the form: v x

= ···, and rearrange the energy equation into the form m = ···. Then combine these simultaneous equations to determine values, first for m and then for v x

X, using the known values of M and u .

is to find a value for the ratio M/m.)

(c) Use your result for v x

to confirm that the relative velocity of approach of the two trucks is the negative of their relative velocity of separation, as expected for an elastic

collision.

Be sure to define all symbols that you use, and show all stages of your working in order to

obtain full marks. You should also clearly show the units at all stages of your calculations.

Question 3

This question is worth 24 marks, and is designed to test your understanding of equilibrium and

torques (Unit 7).

(a) A rigid system is in static equilibrium. State the two conditions that must be satisfied for

this to be the case, both in words and in equations.

A ladder of length 2L and mass M is positioned on level ground and leant against a wall such

that the angle between the ladder and the horizontal is a. The coefficient of static friction

between the ladder and the wall and between the ladder and the ground is µ =0.65. The centre of mass of the ladder is half way along it.

(b) Draw a diagram indicating all the forces acting on the ladder, and state what each force

represents. Also show the direction of the x and y axes you will use. (Hint : there are 2 forces

acting at the top of the ladder, 2 forces acting on the bottom of the ladder and 1 force acting

at its centre.)

(c) For the ladder to be in static equilibrium:

(i) Write down equations for the total x and y components of the 5 forces acting on the

(ii) Consider torques about the centre of the ladder. In which direction (into or out of the

page) does the torque due to each of the 5 forces force act?

(iii) Write down an equation for the sum of the torques about the centre of mass of the

(iv) Use your equation for the torques to derive an expression for tan (a) interms of the

magnitudes of the forces acting.

(d) (i) If the ladder is just on the point of slipping at both the upper and lower ends, what can

you say about the pair of forces acting at each of the top and bottom of the ladder?

(ii) Hence use this information, with the information from (c)(i) and the expression you have

derived in (c)(iv) to calculate the minimum angle that the ladder can form with the ground in order for it not to slip. (Hint : In the expression for tan (a), you will need to write each of the forces in terms of one of the normal reaction forces, which then cancels out, leaving an expression for tan

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