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Mathematics
An Introduction to Mathematics for Engineers: Mechanics
183 SOLVED PROBLEMS
Question: 17.9
Find a unit vector at right angles to the plane containing a = 2i – 6j – 3k and b = 4i + 3j – k. ...
Verified Answer:
The vector product a × b is perpendicular to both ...
Question: 1.8
A juggler throws a ball up in the air with initial speed 5 ms^-1 from a height of 1.2 m. It has a constant acceleration of 10 ms^-2 vertically downwards due to gravity. i) Find the maximum height of the ball above the ground and the time it takes to reach it. At the instant that the ball reaches ...
Verified Answer:
i) In this example it is very important to draw a ...
Question: 5.6
A car of mass 1000 kg, including its driver, is being pushed along a horizontal road by three people as indicated in the diagram. The car is moving in the direction PQ. i) Calculate the total force exerted by the three people in the direction PQ. ii) Calculate the force exerted overall by the three ...
Verified Answer:
i) Resolving in the direction PQ, the components i...
Question: 1.4
A bus leaving a bus stop accelerates at 0.8 ms^-2 for 5s and then travels at a constant speed for 2 minutes before slowing down uniformly at 4 ms^-2 to come to rest at the next bus stop. Calculate i) the constant speed ii) the distance travelled while the bus is accelerating iii) the total distance ...
Verified Answer:
i) The diagram shows the information for the first...
Question: 8.13
i) Find the position of the centre of mass of a uniform hemispherical bowl of thickness 1 cm and inside radius 9 cm. ii) A light handle AB of length 18 cm is attached to a point A on the rim of the bowl in such a way that the line AB passes through the centre of the hemisphere. What angle does AB ...
Verified Answer:
i) Think of the bowl as a solid hemisphere of radi...
Question: 18.9
A heavy ball is retarded by air resistance which is modelled by kv², where v ms^-1 is the speed of the ball and k is a positive constant. When falling under gravity in air, its terminal velocity is V ms^-1. i) Write an expression for the total force on the ball when it is ...
Verified Answer:
i) Assume that the mass of the ball is m and the r...
Question: 14.8
A particle of mass 200 g is attached to a light spring of natural length 40 cm and stiffness 50 Nm^-1. The particle is allowed to hang vertically in equilibrium. i) Find the extension of the spring in this position. The spring is now pulled down 3 cm and released from rest. ii) Find the length of ...
Verified Answer:
i) The diagram shows the relevant lengths, and the...
Question: 14.4
The functions f and g are given by f : x = a sin (ωt + ε) and g : x = a cos (ωt + ε) where ε (epsilon) is a positive constant. For each of these functions: i) Differentiate it with respect to time to find v (or x^.) and show that v² = a²(ω² – x²). ii) Differentiate v with respect to time to ...
Verified Answer:
i) For f: x = a sin(ωt + ε) ⇒ v =
\dot...
Question: 5.5
A skier is being pulled up a smooth 25° dry ski slope by a rope which makes an angle of 35° with the horizontal. The mass of the skier is 75 kg and the tension in the rope is 350 N. Initially the skier is at rest at the bottom of the slope. The slope is smooth. Find the skier’s speed after 5 s and ...
Verified Answer:
In the diagram the skier is modelled as a particle...
Question: 7.8
A rectangular block of mass 3 kg is placed on a slope as shown. The angle α is gradually increased. What happens to the block, given that the coefficient of friction between the block and slope is 0.6? ...
Verified Answer:
Check for possible sliding Figure 7.42 shows the f...
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