Unit 8
10 marks · HardThis question is about compound measures and vector geometry.
A metal block of mass 540 g has a volume of 180 cm³. Calculate the density of the block in g/cm³.
The block is then subjected to a force of 72 N, uniformly distributed across one face of area 30 cm × 6 cm. Calculate the pressure exerted on this face in N/cm².
The force is now applied at two points, A and B. From the centre of the face, the force at A is represented by the vector N, and at B by N. (i) Write the total force vector (A+B), as a column vector.
(ii) Calculate the magnitude of the total force vector. Give your answer to 2 decimal places, including units.
(iii) Find the unit vector in the direction of the total force vector. Give each component to 3 significant figures.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 3.0 g/cm³
(b) 0.4 N/cm²
(c) N
(d) 10.00 N
(e)
Common mistakes: Dividing by wrong units for density or pressure; Sign errors in vector addition; Incorrect vector magnitude calculation
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Substitutes into density = mass / volume formula.
A1 (a) · 1 mark · Both values in correct units and answer correct.
M2 (b) · 1 mark · Correctly calculates area and divides force by area for pressure, in correct units.
A2 (b) · 1 mark · Answer correctly rounded and units included.
M3 (c) · 1 mark · Adds vectors correctly: (15+(-9), -12+20).
A3 (c) · 1 mark · Column vector for total force correct.
M4 (d) · 1 mark · Uses Pythagoras: sqrt(6^2 + 8^2).
A4 (d) · 1 mark · Gives answer correct to 2 decimal places, units included. (follow-through)
M5 (e) · 1 mark · Divides each component of total force by its magnitude. (follow-through)
A5 (e) · 1 mark · Each component rounded to 3 significant figures. (follow-through)
Worked solution
a) Density = mass / volume = 540 g / 180 cm³ = 3.0 g/cm³.
b) Area = 30 cm × 6 cm = 180 cm². Pressure = 72 N ÷ 180 cm² = 0.4 N/cm².
c) Total force = N.
d) Magnitude = N.
e) Unit vector: .