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Question 1:
A car is moving horizontally at a constant speed of ux = 10 ms−1 towards a
cliff edge. The vertical distance from the top of the cliff to the ocean below
is 20 m. The car drives horizontally off the cliff and lands in the ocean
below.
Consider the uniform acceleration equation that describes displacement
as a function of time t,
1
S(t) = ut + at2.
2
a) Explain, using words, why the above equation becomes the equation
stated below when considering only the horizontal displacement of the car,
⇒
Sx(t) = uxt.
[The answer to this part of the question must be typed in own words.]
[3 marks]
b) Calculate the maximum horizontal distance (Sx, max) that the car reaches
when it lands in the ocean (also known as the range).
[3 marks]
Page 2 of 15
Question 2:
Consider an object that has the following acceleration as a function of
time t,
a(t) = 2t + 3.
a) Explain, using words, why we can not use the uniform acceleration
equations to calculate the velocity of this object.
[The answer to this part of the question must be typed in own words.]
[3 marks]
Given that at the time t = 1, the object has a velocity of v(t = 1) = 6 ms−1.
Find an expression for the velocity of this object as a function of time in the
following form,
v(t) = pt2 + qt + k,
where p, q and k are constants.
b) State the numerical values of the constants p, q and k.
[5 marks]
Page 3 of 15
Question 3:
A mass m is oscillating on a spring, there is no resistance to the motion of
the mass. This type of oscillation is described by simple harmonic
motion (SHM) and the differential equation that describes the motion of
this particular mass is stated below,
ẍ(t) = −25x(t).
a) Explain, using words, the meaning of each symbol in the differential
equation. In your answer make sure to state the meaning of the two dots
appearing on the left hand side and the significance of the negative sign
appearing on the right hand side.
[The answer to this part of the question must be typed in own words.]
[4 marks]
b) Calculate the time period for the oscillation of the mass.
[3 marks]
Page 4 of 15
Question 4:
A mass of m = 1.5 Kg is attached to the end of a light inextensible string.
The string then begins to rotate such that the mass travels in a circular
path of radius r and the string makes an angle θ = 300 with the vertical.
This type of arrangement is called a conical pendulum. The force T
represents the tension in the string. See diagram below.
a) Resolve the forces vertically to calculate the magnitude of the tension T
in the string.
[3 marks]
b) Explain, using words, how you would find the magnitude of the
circular force keeping the mass moving on the circular path. In your answer
make reference to what is providing the circular force and the direction of
the circular force.
[The answer to this part of the question must be typed in own words.]
[3 marks]
c) Therefore, calculate the circular acceleration that the mass experiences
whilst moving in the circular path. State which one of Newton’s laws you
have used in the calculation of the acceleration.
[3 marks]
Page 5 of 15
Question 5:
A ball is initially at rest and at a height of 1.8 m above a flat horizontal
surface. The ball is then released and fulls under the influence of gravity
vertically downwards towards the surface. The coefficient of restitution
between the ball and surface is e = 0.3.
a) Explain, using words, how the coefficient of restitution between the ball
and the surface can be found by doing an experiment. (Hint: think of the
parameters in the mathematical definition of the coefficient of restitution
and how these parameters can be found).
[The answer to this part of the question must be typed in own words.]
[3 marks]
b) By using the uniform acceleration equations, calculate the maximum
vertical height that the ball reaches after one bounce off the surface.
Assume that the surface remains stationary throughout.
[4 marks]
Page 6 of 15
Question 6:
Consider a horizontal bar that is on a pivot. At one end of the bar is a mass
M at a distance of d1 = 0.3 m from the pivot point. At the other end of the
bar is a mass m at a distance of d2 from the pivot point. The pivot is in the
position such that the bar remains horizontal. See diagram.
𝒅𝟏
𝒅𝟐
Consider the scenario that the mass M is larger than the mass m, such
that M > m.
a) Explain, using words, how by considering the moment at the pivot point
caused by each mass we can deduce that the distance d2 must be larger
than the distance d1 such that d2 > d1 for the bar to remain horizontal.
[The answer to this part of the question must be typed in own words.]
[3 marks]
b) If the mass M = 4m, find the corresponding distance d2.
[4 marks]
Page 7 of 15
Question 7:
Consider two lamina rectangles A and B which have the corresponding
masses mA = 4 Kg and mB = 2 Kg respectively. Assume each rectangle to
have a uniform mass density. Rectangle A has dimensions 2 m by 10 m
and rectangle B has dimensions 4 m by 2 m.
The two rectangles are attached to each other and the point O represents
the end of the horizontal line that passes through the centre of gravity of
the entire shape. See diagram below.
𝟏𝒎
𝑩
𝑨
𝟐𝒎
𝟒𝒎
𝒐
𝟏𝟎𝒎
𝟐𝒎
a) Explain, using words, how by considering lines of symmetry of each
rectangle we are able to find the center of gravity for each rectangle
individually. In your answer make reference to the uniform distribution of
mass in each rectangle.
[The answer to this part of the question must be typed in own words.]
[3 marks]
b) Calculate the horizontal distance of the centre of mass of the combined
shape from the point O.
[3 marks]
Page 8 of 15
Question 8:
A ball is launched from the ground at an angle of θ = 60o and an initial
velocity of u = 50 ms−1. The ball then lands a horizontal distance away
from where it started at the same initial vertical height. See the diagram
below.
𝜗
Consider the uniform acceleration equation that describes the vertical
displacement of the ball as a function of time t,
1
gt2.
2
Where g is the acceleration due to gravity taken to be g = 9.8 ms−2.
Sy (t) = u yt −
a) Use this equation to calculate the time of flight (T ) of the ball.
[3 marks]
b) Explain, using words, why there are two time solutions to the
equation above. What does the other time solution represent physically?
[The answer to this part of the question must be typed in own words.]
[3 marks]
Page 9 of 15
Question 9:
A mass of m = 5 Kg is moving on a straight line. Its displacement from the
origin as a function of time t is given by,
S(t) = 9t − t3.
a) Calculate the kinetic energy of the mass at t = 1 s.
[4 marks]
b) Explain, using words, how to find the times that correspond to the mass
passing through the origin. What would be the physical meaning of a
negative time solution?
[The answer to this part of the question must be typed in own words.]
[4 marks]
Page 10 of 15
Question 10:
A mass is oscillating with simple harmonic motion (SHM). The time period
of the oscillation is T = 1.2 s and the amplitude of the mass is a = 0.36 m.
The velocity of the mass as a function of displacement from the
equilibrium position is given by,
√
v(x) = ω a2 − x2.
a) Calculate the velocity of the mass when it is passing through the
equilibrium position.
[3 marks]
b) Explain, using words, why the velocity of the mass is at its minimum
when the mass is at maximum displacement from the equilibrium
position. State the value of the minimum velocity. (Hint: consider the
equation stated above).
[The answer to this part of the question must be typed in own words.]
[4 marks]
Page 11 of 15
Question 11:
A cube of mass m = 1.5 Kg is stationary on a horizontal rough disk at a
radius of r = 0.15 m from the centre of the disk. The disk has a coefficient
of friction of µ = 0.8. See diagram.
𝝎
𝝁
The disk begins to rotate at a constant angular speed of ω = 4 s−1.
a) Calculate the magnitude of the circular force that the cube experiences
whilst the disk is rotating.
[3 marks]
b) Therefore, what is the magnitude of the frictional force between the
mass and the disk?
[1 mark]
The angular speed of the disk is increased. The maximum angular speed
for which the mass remains on the rotating disk is found to be
ωmax = 7.23 s−1. If the angular speed of the disk is increased above ωmax
then the mass slides off the rotating disk.
c) Explain, using words, why this maximum angular speed of the disk
exists. (Hint: think about the force that is providing the circular force).
[The answer to this part of the question must be typed in own words.]
[5 marks]
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Question 12:
An object A has a mass mA = 2 Kg and initial velocity vA = 3 ms−1.
Another object B has a mass mB = 3 Kg and initial velocity vB = 1 ms−1.
The objects are initially a horizontal distance d apart from each other. Both
objects are moving horizontally on a smooth surface in a straight line and
in the same direction. See diagram.
𝒗𝑨
𝒗𝑩
𝒅
Both of the objects A and B collide and merge into one single object C (the
objects coalesce) which continues to move in the same direction at the
final velocity vC.
a) Explain, using words, how we can use the principle of the
conservation of momentum to find the final velocity v C. State, using words,
a condition that must be assumed when applying this conservation law.
[The answer to this part of the question must be typed in own words.]
[4 marks]
b) Use momentum conservation to calculate the final velocity v C.
[3 marks]
Page 13 of 15
Question 13:
A force of magnitude | P̄| = 10 N is acting on a rigid horizontal rod at an
angle θ = 30o. The opposite end of the rod is fixed and is labelled A, this is
the point which the rod can rotate around. The distance between the point
A and the point at which the force p̄ acts is d = 3 m. See diagram.
𝑨
𝒅
𝜗
𝑝
a) Calculate the moment at the point A. In your answer make sure to
indicate the sense of rotation.
[3 marks]
Now consider another force F̄ acting on the rod. This additional force
is passing through the point A on the rod.
b) Explain, using words, what the effect this additional force will have when
considering the moment at the point A. In your answer make use of the
phrase line of action.
[The answer to this part of the question must be typed in own words.]
[4 marks]
Page 14 of 15
Question 14:
Three masses m1 = 4 Kg, m2 = 3 Kg and m3 = 2 Kg are situated on a
coordinate grid. The coordinates for each of the masses respectively are
(0, 8), (2, 0) and (6, 2). See diagram below.
𝒎𝟏
𝒎𝟑
𝒎𝟐
The center of gravity of these three masses is denoted by the point G
which has the coordinates (x̄, ȳ).
a) Find the numerical values of the coordinates of the point G.
[6 marks]
END OF EXAM.
Page 15 of 15

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