MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics

Physics 8.01 Fall Term 2011

Exam 1 Solutions Problem 1 of 4 (25 points): Cars on the road

At time t = 0 , two cars are next to each other, traveling in the same direction with initial speed v0 . Car 1 moves at constant speed, while car 2 accelerates according to

a2 = ! " #t, for 0 < t < t f , where ! and ! are positive constants. At time t = t f , car 2 has traveled twice as far and has twice the speed of car 1. Find ! and ! in terms of v0 and t f . Solutions:

Problem 2 of 4 (25 points): Forces and vectors

A block of mass M is lying stationary on an inclined plane with angle ! with respect to the horizontal. There is static friction between the block and the inclined plane with coefficient µs . The angle ! is less than the maximum angle such that the block does not slip. The magnitude of the acceleration of gravity is g in the downward vertical direction.

a) Draw a free-body (force) diagram for the block. Clearly identify your forces with whatever symbols you find appropriate.

b) Suppose we define two separate coordinate systems. In the first system, i points

horizontally to the right and j points vertically upward. In the second system, b points down along the inclined plane, and c points upwardly normal (perpendicular) to the plane. For each of the two coordinate systems, make a vector decomposition of each of your forces in your free-body (force) diagram. Then write down an expression for the total vector force on the block in terms of the components (your choice of symbols) and unit vectors.

c) Suppose we could vary the angle ! of the inclined plane. What is the maximum

angle ! for which the block will remain stationary and not slide? (You may use any coordinate system to solve this problem.)

Problem 3 of 4 (26 points): Blocks-pulley system

Block 1 and block 2, with masses m1 and m2 such that m1 << m2 ( m1 is much less than

m2 ), are connected by a massless inextensible string wrapped around a massless ideal pulley. The pulley is rigidly connected to the top of an inclined plane which makes an angle ! with the horizontal, as shown in the figure above. Block 2 slides down the inclined plane, and block 1 slides up block 2. The coefficient of kinetic friction between the blocks is µk . The surface between the lower block and the inclined plane is frictionless. Gravity is directed vertically downward with acceleration g .

a) Draw free-body (force) diagrams for the two blocks. Note: You should clearly define your choice of coordinate system and unit vectors for each object.

b) Solve for the acceleration of each block in terms of m1 , m2 , µk , ! , and g .

Solution

Problem 4 of 4: (24 points) Concept Questions Circle the correct answer for the following four parts. Part A (6 points): Joe is traveling in a train that is moving at a constant velocity. He leans out the window and throws a package to Susan who is standing beside the tracks below the train window. Susan catches the package, which seems to her to fall vertically into her hands. Which of the following did Joe do?

1. Joe threw the package straight down from the window of the moving train. 2. Joe threw the package toward the front of the moving train.

3. Joe threw the package toward the back of the moving train.

4. Not enough information given to decide.

Answer 3

Part B (6 points): At t = 0 , an object is located at the origin. The object moves along the x -axis with the x -component of its velocity given by

vx (t) = vx0 ! "t 3 , where ! is a positive constant and vx0 > 0 is the x -component of the velocity at t = 0 . Which of the following statements is true?

1. For t > 0 , the position of the object will always be positive. 2. For t > 0 , the position of the object will always be negative.

3. At some later time t1 > 0 , the object will stop and reverse direction. 4. At some later time t1 > 0 , the object will stop and remain at rest.

5. The x -component of the acceleration of the object is increasing as a function of

time.

6. The x -component of the acceleration of the object starts off with a positive non-zero value at t = 0 and then decreases as a function of time.

Answer 3

Part C (6 points): Two identical stones slide off a frictionless horizontal tabletop at the same instant and fall to ground which is also horizontal (parallel to the tabletop). Just before sliding off, one is sliding at a speed v0 and the other one is sliding at a speed 10 v0 . Which stone hits the ground first?

1. The stone with initial speed v0 . 2. The stone with initial speed 10 v0 .

3. Both land at the same time.

5. Not enough information given to decide.

Answer 3

Part D (6 points): Block 1 of mass m1 is next to block 2 of mass m2 , with m2 > m1 . The two blocks are on a horizontal frictionless surface. If a horizontal force

!F1 is applied to block 1 and

!F2 is

the horizontal contact force acting on block 2, then which of the following statements is true?

1. !F1 >

!F2

2. !F1 <

!F2

3. !F1 =

!F2

4. Not enough information given to decide.

Answer 1