Science
PHYS 1 University of Liverpool Physics Unit Vector Problem Set

PHYS 1

University of Liverpool

PHYS

Question Description

-All questions must be solved CORRECTLY

-show all working out for each question

-DO NOT USE outside sources to answer any question

-can be answered through hand written work and then sent clearly back

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Question 1: Consider the vector r¯ stated below, r¯ = −6ˆi + 3ˆj. a) Find the unit vector rˆ of the vector stated above. [4 marks] b) Explain, using words, how you would find an expression for the vector R¯ that is parallel to the vector r¯ but has a length |R¯|. [The answer to this part of the question must be typed in own words.] [3 marks] 2 Question 2: Two vectors a¯ and ¯b are expressed below in the unit vector notation, a¯ = 4ˆi − 2ˆj + 8kˆ, ¯b = 7ˆi + 1kˆ. a) Calculate the angle θ between the two vectors a¯ and ¯b. [4 marks] b) What can you infer about the scalar product ˆi · ˆj. Make sure you justify your answer using words. [The answer to this part of the question must be typed in own words.] [3 marks] 3 Question 3: A ball falls off a ledge and moves vertically downwards under the influence of gravity towards the ground. The ledge is positioned at a height of h = 7.90 m above the ground. a) Calculate the time t required for the ball to reach the ground. [4 marks] b) If the initial height of the ledge was doubled (h → 2h), does the amount of time required for the ball to fall to the ground also double? Explain your answer using words. Comment on the factor increase in the time taken to fall a height of 2h when compared to h. [The answer to this part of the question must be typed in own words.] [4 marks] 4 Question 4: An object starts at an initial position and is moving horizontally on a straight line with an initial velocity of u = 11ˆi ms−1. The object experiences an acceleration in the opposite direction to its initial motion, the magnitude of this acceleration is |a¯| = 3.50 ms−2. a) Express the acceleration as a vector in terms of the unit vector ˆi. Explain, using words, the sign convention and what it represents. [The answer to this part of the question must be typed in own words.] [4 marks] b) Calculate the distance between the initial position of the object and the position at which the object comes to rest. [4 marks] 5 Question 5: A mass of m = 3 Kg is positioned on a smooth incline at an angle of elevation of θ = 12o above the horizontal. The mass is at a distance d from the bottom of the incline. see diagram below. 𝑚 d 𝜗 a) Calculate the acceleration of the mass down the incline. [4 marks] b) Calculate the reaction force that the surface exerts on the mass. [3 marks] Now consider the surface of the incline to be rough such that the coefficient of friction is non-zero µ = 0. c) Explain, using words, how a rough surface will affect the acceleration experienced by the mass. [The answer to this part of the question must be typed in own words.] [3 marks] d) Calculate the value of the coefficient of friction µ needed for the mass to be in limiting equilibrium. [3 marks] 6 Question 6: A box of mass m = 17 Kg is on a smooth horizontal surface with a force F¯ acting on it at an angle θ = 24o above the horizontal. See diagram. 𝐹 𝜗 𝑚 a) If the horizontal acceleration of the box is ax = 6 ms−2, find the magnitude of the pulling force F¯. [4 marks] b) Calculate the corresponding reaction force R that the horizontal surface exerts on the box. [3 marks] c) Explain, using words, how your consideration of Newton’s third law helped you to calculate the reaction force that the surface exerts on the mass in part b). [The answer to this part of the question must be typed in own words.] [3 marks] Consider the surface to now be rough with a coefficient of friction given by µ = 0.40. d) Calculate the new horizontal acceleration of the box. [3 marks] 7 Question 7: A mass m1 = 15 Kg is initially at rest on a smooth horizontal surface. The mass m1 is attached to a light in-extensible string which runs over a smooth pulley and is attached to another mass m2 = 7 Kg that hangs ver- tically downwards. See diagram. 𝑚1 𝑚2 a) Use Newton’s second law to find an equation of motion for each of the masses. [4 marks] b) Solve the two equations of motion simultaneously to find the the value of the tension T in the string and the acceleration a of the masses. [4 marks] c) Calculate the time required for the mass m2 to fall a vertical distance of 1.20 m given that the mass was initially at rest. [3 marks] 8 Question 8: A mass m2 = 5 Kg is attached to a light in-extensible string that passes over a smooth fixed pulley and under another smooth pulley of mass m1 = 1 Kg. Both masses are free to move vertically and initially the system is at rest. When the masses are in motion the mass m1 accelerates vertically upwards whilst the mass m2 accelerates vertically downwards. See diagram. 𝑚1 𝑚2 a) Use Newton’s second law of motion to find an equation of motion for each of the masses. [4 marks] b) Solve the two equations of motion simultaneously to calculate the acceleration of the mass m2. [4 marks] c) Given that the masses were initially at rest, calculate the velocity of the mass m2 after it has been in motion for t = 4.00 s. [3 marks] 9 Question 9: A mass of m = 3 Kg is sliding in a straight line across a rough horizontal surface with a coefficient of friction µ = 0.50. The initial velocity of the mass is v = 6 ms−1 and it eventually comes to rest at a distance d from its initial position. a) By considering work-energy equivalence, calculate the work done by friction in bringing the mass to rest. [3 marks] The work done by a constant force is given by, W = F d, where W is the work done, F is the magnitude of the constant force and d is the distance moved. b) Explain, using words, why the constant force due to friction in this case is equal to Fmax, the maximum frictional force that can be provided by the surface. [The answer to this part of the question must be typed in own words.] [4 marks] c) Use energy conservation to calculate the distance d. [4 marks] 10 Question 10: A mass of m = 15 Kg is sliding across a smooth horizontal surface with a velocity v = 8 ms−1. The mass collides directly with a horizontal spring and causes the spring to compress until the mass comes to rest. The value of the spring constant is k = 227 Nm−1. a) Explain, using words, how energy conservation can be used to calculate the compression of the spring. [The answer to this part of the question must be typed in own words.] [4 marks] b) Calculate the initial kinetic energy of the mass. [2 marks] The elastic energy stored in a spring is given by, 1 Eelastic = kx 2, 2 where Eelastic is the elastic energy, k is the spring constant and x is the compression/extension of the spring. c) Calculate the compression of the spring needed to bring the mass to rest. [5 marks] . 11 ...
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