Module 1 — Problem Set 1

Before you start

There are no answers on this page, and that is deliberate. If you cannot tell whether your answer is right, that is worth knowing — it is a different problem from getting it wrong, and only one of the two is fixed by looking something up.

Each problem names the Foundation it tests. If one defeats you, that is where to reread rather than guess. Several of these are close to worked examples published elsewhere on the site, and finding those is part of the exercise.

This is the easier of the sets. Work through it before the harder ones.

A quantity Q is thought to depend on a mass m, a speed v and a length L as Q = C m^av^bL^c, where C is a dimensionless constant. The units of Q are kg m² s⁻².

(a) Find a, b and c.

(b) Another student proposes Q = C m v²L⁻¹. Decide whether this can represent Q, without computing any numerical value.

A particle undergoes three successive displacements in a plane: d₁ = 3î m, then d₂ = 4ĵ m, then d₃ = −3î m.

(a) Find the total distance travelled.

(b) Find the final displacement vector and its magnitude.

(c) A student argues that distance and displacement must always be equal, because both are measured in metres. Identify precisely what is wrong with that reasoning.

A point P has Cartesian coordinates (−4, 4√3) m.

(a) Find its polar coordinates, with r ≥ 0 and 0 ≤ θ < 2π.

(b) Write the unit vectors r̂ and θ̂ at P in terms of î and ĵ.

(c) A point Q is obtained by rotating P anticlockwise about the origin through 90°. Give its Cartesian coordinates and its polar angle.

Three coplanar forces act at a point: F₁ = (6î + 8ĵ) N, F₂ = (−4î + 3ĵ) N, and F₃ = (Aî + Bĵ) N. The net force is zero.

(a) Find A and B.

(b) Find the magnitude of F₃ and its direction measured anticlockwise from the positive x-axis. Take care over the quadrant.

A force F₁ of magnitude 10 N acts in the first quadrant at an angle θ above the positive x-axis, and its x-component is twice its y-component.

(a) Find θ and both components.

(b) A second force of 5 N acts vertically downward. Find the resultant of the two and its magnitude.

A ring is held in equilibrium by three tensions: T₁ of magnitude T₁ directed along (−î − ĵ)/√2, T₂ of magnitude T₂ along +î, and T₃ = 20ĵ N.

(a) Write the equilibrium condition as two scalar equations and solve for T₁ and T₂.

(b) If T₂ cannot exceed 30 N, find the largest possible magnitude of T₃ with the directions unchanged.

A constant force F = (6î + 8ĵ) N acts on a particle during a displacement of magnitude 10 m whose direction is not specified. Write the displacement as 10d̂.

(a) Find the direction d̂ that makes the work greatest, and that greatest value.

(b) Find every direction for which the work is zero.

(c) Find the direction that makes the work least, and that value.

Given A = 2î + 3ĵ + 6k̂ and B = î − 2ĵ + 2k̂:

(a) Find the scalar projection of A onto B.

(b) Split A into a part parallel to B and a part perpendicular to it.

(c) Verify both that the two parts are perpendicular and that their squared magnitudes add to |A|². Say why each check is worth doing.

A force F = 4ĵ N is applied, in turn, at the point P = (2, 0, 0) m and at the point Q = (0, 2, 0) m. Torque about the origin is τ = r × F.

(a) Find τ in each case.

(b) Compare the two magnitudes and explain the geometric reason for the difference. One of them may surprise you.

A force of 20 N is applied at the tip of a rod of length 0.5 m pivoted at the origin. The angle φ between the rod and the force can be set anywhere in [0, π].

(a) Find the φ that maximises the torque, and its value.

(b) Find every φ for which the torque is zero.

(c) Both answers can be obtained without calculation. Explain how.

A particle moves in a straight line. Its velocity–time graph consists of straight segments joining the points (0, 2), (2, 6), (5, 0) and (7, −4), with time in seconds and velocity in m/s. Sketch it before answering.

(a) Find the acceleration on each segment.

(b) Find the net displacement from t = 0 to t = 7 s.

(c) State when the particle is moving in the negative direction.

(d) Find the total distance travelled.

A particle moves along the x-axis with x(t) = 2t³ − 9t² + 12t + 1 metres, for t ≥ 0.

(a) Find v(t) and a(t).

(b) Find every instant at which the particle is momentarily at rest.

(c) Find the acceleration at each of those instants, and say what the sign of each tells you. Do not use the constant-acceleration equations anywhere.

Power is delivered to a system at a rate P(t) watts given by 10t for 0 ≤ t < 2 s, then a constant 20 for 2 ≤ t < 5 s, then 20e^−(t−5) for t ≥ 5 s.

(a) Sketch P(t) up to t = 8 s.

(b) Find the total energy delivered, integrating piece by piece out to infinity.

(c) Find the instant at which exactly half the total energy has been delivered.

The pendulum equation is made solvable by replacing sin θ with θ, in radians.

(a) Find the percentage error |(sin θ − θ) / sin θ| × 100 at θ = 30°.

(b) Decide whether the approximation is acceptable at 30° if the tolerance is 5%, 1%, or 0.1%.

(c) Explain why whether an approximation is valid depends on the application and not only on the function.

When you have finished

Go back over the ones you found hard and ask what kind of difficulty it was. Not knowing the method is one thing, and rereading the Foundation fixes it. Knowing the method and setting the problem up wrongly is another, and rereading will not help — that is a matter of practice at deciding what to do first.

Four of these have a part that can be answered with no calculation at all, if you see the structure. Finding which four is worth more than finishing quickly.

Covers: Foundations 1–10  ·  Type: Problems, no answers  ·  Difficulty: Beginner to Intermediate