Exercises on Newton s Laws of Motion

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Dwain Weaver
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1 Exercises on Newton s Laws of Motion Problems created by: Raditya 1. A pendulum is hanging on a ceiling of a plane which is initially at rest. When the plane prepares to take off, it accelerates with a constant acceleration until its speed reaches 30 m/s. It is known that during this period, the pendulum makes an angle 45 o with respect to vertical line. Find the distance traveled by the plane until it takes off. [Answer: x = m 2. A block of mass m is at rest on a rough surface which makes an angle θ with respect to horizontal. The coefficient of static friction is µ s, but it is too small for the friction alone to be able to prevent the block from sliding down. If an external force F is to be applied to the block, as shown below, find its range of values in order to keep the block in equilibrium. 3. Look at the figure below, There is no friction between block C and the ground, but there is friction between block C and D. Also, assume that all the strings are massless and frictionless. Find the minimum coefficient of static friction between block C and D such that all the blocks move with the same acceleration. [Answer: µ s,min = (m B+m C )m A +(m A +m D )m B m D (m A +m B +m C +m D ) 4. Look at the figure below, 1

2 Assume that m B > m A and that the pulley is massless. There is no friction between block B and the ground as well as between the pulley and the ground, while the coefficient of static and kinetic friction between block A and block B is µ s and µ k respectively. a. Find the maximum value of F such that the two blocks move with the same acceleration? [Answer: F max = 2m A gµ s(m A +m B ) m B m A b. Suppose F is greater than the maximum value found in the first part. Find the acceleration of each block. [Answer: a A = F 2m Agµ k 2m A, a B = F +2m Agµ k 2m B c. Suppose F is the same as in part 2, find the acceleration of the pulley. [Answer: a P = a A+a B 2 5. [CH Consider a generalization of the previous question as shown below, There is an infinite number of mass each having the same mass m. When the horizontal force F is applied, find the acceleration of the pulley that is being pulled by the force. [Answer: a = F 3m 6. [H Consider the following figure, Assume that m 2 > m 1 and that the string is massless. However, there is friction between the pulley and the string. Find the minimum coefficient of static friction between the string and the pulley so that both masses are not accelerating. [Answer: µ s,min = ln ( m2 m 1 ) π 7. Similar as the previous question, but now the string is frictionless as well as as massless. However, we now pull the pulley by an upward force F. Find the relation between m 1 and m 2 in order for the pulley to have zero acceleration. [Answer: m 1 = F m2 4m 2g F 2

3 8. [CH Consider another infinite pulley system below, where each mass m is the same. The main pulley is pulled by an upward force F. Find the maximum value of m in order for the main pulley to not accelerate downward. [Answer: m max = F 3g 9. Look at the figure below, The coefficient of static and kinetic friction between the two blocks are µ s and µ k, and there is no friction between the blocks and the ground. a. Assuming θ is fixed, Find the range of values of F so that block m B doesn t go upward or downward. [Answer: m B g cos(θ)+ m A µs sin(θ) m A +m B F m B g cos(θ) m A µs sin(θ) m A +m B 3

4 b. Let θ = 90 o now, so that the force F is horizontal. However, now there is another force f = m Bg µ s pushing block m A to the right. Find the minimum value of F so that block m B doesn t fall. [Answer: F min = m Bg µ s 10. Consider a system of blocks, all with the same mass m, arranged in the manner as shown in the following figure. There is no friction between the blocks and the ground, but there is friction between each pair of blocks with the same coefficient of static and kinetic friction µ s = µ k = 1 4. Find the range of values for θ such that: a. All the blocks move with the same acceleration. [Answer: 0 o θ 26.6 o b. Block C slips, but the rest of the blocks still move together with the same acceleration. [Answer: 26.6 o θ 32.6 o 11. Consider a system of two blocks pushed by a certain force F against a vertical wall, as shown below. There is no friction between m 1 and the wall, but there is friction between m 2 and m 1, with coefficient of static and kinetic friction µ s and µ k respectively. [ a. Find the minimum angle θ so that both blocks move with the same acceleration. [Answer: θ = arctan m 1 (m 1+m 2)µ s b. Suppose that θ is less than the one found in part a. Find F such that the acceleration of m 1 and m 2 are equal in magnitude but opposite in direction. [Answer: F = 2m 1m 2g m 1 cos(θ)+(m 2 m 1) sin(θ)µ k. 12. A certain UFO takes the shape of an inverted bowl (inverted half-sphere without the base surface) with radius R. As it flies around the sky, it rotates about its axis with a period of revolution of T = π. Find the minimum coefficient of static friction µ s of the UFO material used so that the aliens are able to stand on the wall of the UFO up to an angle of θ > arctan ( 1 4) with respect to its axis (vertical line) as it flies around the sky under the influence of Earth s gravity. [Answer: µ s = 4 cos(θ)+sin(θ) 4 sin(θ) cos(θ) R g 4

5 13. A block m 2 is on top of another block m 1, which is placed along an inclined ground, as shown in the following figure. Assuming that there is no friction between m 1 and the inclined surface, but there is friction between m 1 and m 2 such that both blocks move together with the same acceleration, find this friction force. [Answer: f = m 2 g cos(θ 1 ) sin(θ 2 ) 5

Webreview practice test. Forces (again)

Webreview practice test. Forces (again) Please do not write on test. ID A Webreview 4.3 - practice test. Forces (again) Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 5.0-kg mass is suspended