Question 1 (6 marks)
a) Newton’s universal law of gravitation describes the force of gravity acting on two masses. The
correct equation is,
Fg = G
m1m2
r
2
.
Using dimensional analysis, determine the dimensions and SI units for the gravitational constant
G. Here, Fg is a force, m1 and m2 are masses and r is a distance. (2 marks)
b) Someone then tells you that the equation for gravitational potential energy U (measured in
Joules) is,
U = −G
m1m2
r
3
.
Using dimensional analysis, determine if they are correct. (1 mark)
c) An experiment determined that the time for a star to orbit a black hole T in a circular orbit
depends on the distance from the black hole to the star r, the gravitational constant G and the
mass of the black hole m. That is,
T = CrαG
βmγ
where C is a dimensionless constant. Using only dimensional analysis, determine what the
exponents α, β and γ must be for this equation to be correct. (3 marks)
Question 2 (7 marks)
Answer the following given the two vectors written in standard position.
A⃗ = 4m/s @ 220o B⃗ = 7m/s @ 300o
a) Draw a diagram for A⃗ and write A⃗ in terms of ˆi and ˆj (answer to 2 decimal places). (2 marks)
b) Draw a diagram for B⃗ and write B⃗ in terms of ˆi and ˆj (answer to 2 decimal places). (2 marks)
c) Determine the angle θ between A⃗ and B⃗ using the dot product. Then draw a diagram and use
geometry to verify your answer. (3 marks)
Question 3 (8 marks)
Answer the following given the two vectors (measured in meters).
A⃗ = −4ˆi + 10ˆj + ˆk B⃗ = 5ˆi + 5ˆk
a) Using only the dot product (no cross products) determine a unit vector Cˆ that is perpendicular
to both A⃗ and B⃗ . I recommend checking that your answer is indeed perpendicular to both A⃗ and
B⃗ by taking a dot product with each. (6 marks)
b) Determine the angle ϕ (Greek letter for F, pronounced ’f-eye’) between Cˆ and the z-axis (answer
to 2 decimal places). (2 marks)
