2013 F = Ma

Part of F = Ma

Subcontests

(23)

1

2013 F=ma Problem 25

A box with weight

W

will slide down a

30^\circ

incline at constant speed under the influence of gravity and friction alone. If instead a horizontal force

P

is applied to the box, the box can be made to move up the ramp at constant speed. What is the magnitude of

P

?

<span class='latex-bold'>(A) </span> P = W/2 \\ <span class='latex-bold'>(B) </span> P = 2W/\sqrt{3}\\ <span class='latex-bold'>(C) </span> P = W\\ <span class='latex-bold'>(D) </span> P = \sqrt{3}W \\ <span class='latex-bold'>(E) </span> P = 2W

1

2013 F=ma Problem 24

A man with mass

m

jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance

H

at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant

k

, and stretches from an original length of

L_0

to a final length

L = L_0 + h

. The maximum tension in the Bungee cord is four times the weight of the man.
Find the maximum extension of the bungee cord

h

.

<span class='latex-bold'>(A) </span> h = \frac{1}{2}H \\ \\ <span class='latex-bold'>(B) </span> h = \frac{1}{4}H\\ \\ <span class='latex-bold'>(C) </span> h = \frac{1}{5}H\\ \\ <span class='latex-bold'>(D) </span> h = \frac{2}{5}H\\ \\ <span class='latex-bold'>(E) </span> h = \frac{1}{8}H

1

2013 F=ma Problem 23

A man with mass

m

jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance

H

at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant

k

, and stretches from an original length of

L_0

to a final length

L = L_0 + h

. The maximum tension in the Bungee cord is four times the weight of the man.
Determine the spring constant

k

.

<span class='latex-bold'>(A) </span> \frac{mg}{h}\\ \\ <span class='latex-bold'>(B) </span> \frac{2mg}{h}\\ \\ <span class='latex-bold'>(C) </span> \frac{mg}{H}\\ \\ <span class='latex-bold'>(D) </span> \frac{4mg}{H}\\ \\ <span class='latex-bold'>(E) </span> \frac{8mg}{H}

1

2013 F=ma Problem 17

Two small, equal masses are attached by a lightweight rod. This object orbits a planet; the length of the rod is smaller than the radius of the orbit, but not negligible. The rod rotates about its axis in such a way that it remains vertical with respect to the planet. Is there a force in the rod? If so, tension or compression? Is the equlibrium stable, unstable, or neutral wrt small perturbations in the vertical angle of the rod?
(A) There is no force in the rod; the equilibrium is neutral. (B) The rod is in tension; the equilibrium is stable. (C) The rod is in compression; the equilibrium is stable. (D) The rod is in tension; the equilibrium is unstable. (E) The rod is in compression; the equilibrium is unstable.

1

2013 F=ma Problem 21

A simple pendulum experiment is constructed from a point mass

m

attached to a pivot by a massless rod of length

L

in a constant gravitational field. The rod is released from an angle

\theta_0 < \frac{\pi}{2}

at rest and the period of motion is found to be

T_0

. Ignore air resistance and friction.
The experiment is repeated with a new pendulum with a rod of length

4L

, using the same angle

\theta_0

, and the period of motion is found to be

T

. Which of the following statements is correct?

<span class='latex-bold'>(A) </span> T = 2T_0 \text{ regardless of the value of } \theta_0\\ <span class='latex-bold'>(B) </span> T > 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ <span class='latex-bold'>(C) </span> T < 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ <span class='latex-bold'>(D) </span> T < 2T_0 \text{ with some values of } \theta_0 \text{ and } T > 2T_0 \text{ for other values of } \theta_0\\ <span class='latex-bold'>(E) </span> T \text{ and } T_0 \text{ are not defined because the motion is not periodic unless } \theta_0 \ll 1

1

2013 F=ma Problem 20

A simple pendulum experiment is constructed from a point mass

m

attached to a pivot by a massless rod of length

L

in a constant gravitational field. The rod is released from an angle

\theta_0 < \frac{\pi}{2}

at rest and the period of motion is found to be

T_0

. Ignore air resistance and friction.
What is the maximum value of the tension in the rod?

<span class='latex-bold'>(A) </span> mg\\ <span class='latex-bold'>(B) </span> 2mg\\ <span class='latex-bold'>(C) </span> mL\theta_0/T_0^2\\ <span class='latex-bold'>(D) </span> mg \sin \theta_0\\ <span class='latex-bold'>(E) </span> mg(3 - 2 \cos \theta_0)

1

2013 F=ma Problem 19

A simple pendulum experiment is constructed from a point mass

m

attached to a pivot by a massless rod of length

L

in a constant gravitational field. The rod is released from an angle

\theta_0 < \frac{\pi}{2}

at rest and the period of motion is found to be

T_0

. Ignore air resistance and friction.
At what angle \theta_g during the swing is the tension in the rod the greatest?
(A) \text{The tension is greatest at } \theta_g = \theta_0.\\ (B) \text{The tension is greatest at }\theta_g = 0.\\ (C) \text{The tension is greatest at an angle } 0 < \theta_g < \theta_0.\\ (D) \text{The tension is constant.}\\ (E) \text{None of the above is true for all values of } \theta_0 \text{ with } 0 < \theta_{0} < \frac{\pi}{2}

1

2013 F=ma Problem 16

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume \rho_0, and occupy a region of radius

r_0

. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.
How much time passes until the cloud collapses fully? (The constant

0.5427

\sqrt{\frac{3 \pi}{32}}

<span class='latex-bold'>(A) </span> \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ <span class='latex-bold'>(B) </span> \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ <span class='latex-bold'>(C) </span> \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ <span class='latex-bold'>(D) </span> \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ <span class='latex-bold'>(E) </span> \frac{0.5427}{\sqrt{G \rho_0}}r_0

1

2013 F=ma Problem 15

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is

5/9

that of water. At equilibrium, what portion of the rod is above water?

<span class='latex-bold'>(A) </span> 0.25\\ <span class='latex-bold'>(B) </span> 0.33\\ <span class='latex-bold'>(C) </span> 0.5\\ <span class='latex-bold'>(D) </span> 0.67\\ <span class='latex-bold'>(E) </span> 0.75

1

2013 F=ma Problem 14

m

12 \text{ m/s}

to the right collides elastically with a cart of mass

4.0 \text{ kg}

that is originally at rest. After the collision, the cart of mass

m

moves to the left with a velocity of

6.0 \text{ m/s}

. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass (

v_{\text{cm}}

) of the two carts before the collision?

<span class='latex-bold'>(A) </span> v_{\text{cm}} = 2.0 \text{ m/s}\\ <span class='latex-bold'>(B) </span> v_{\text{cm}}=3.0 \text{ m/s}\\ <span class='latex-bold'>(C) </span> v_{\text{cm}}=6.0 \text{ m/s}\\ <span class='latex-bold'>(D) </span> v_{\text{cm}}=9.0 \text{ m/s}\\ <span class='latex-bold'>(E) </span> v_{\text{cm}}=18.0 \text{ m/s}

1

2013 F=ma Problem 13

There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part of the satellites of Saturn, we need to know the relation between the velocity

v

of each layer in the ring and the distance

R

of the layer to the center of Saturn. Which of the following statements is correct?

<span class='latex-bold'>(A) </span>

v \propto R

, then the layer is part of Saturn.

<span class='latex-bold'>(B) </span>

v^2 \propto R

, then the layer is part of the satellites of Saturn.

<span class='latex-bold'>(C) </span>

v \propto 1/R

, then the layer is part of Saturn.

<span class='latex-bold'>(D) </span>

v^2 \propto 1/R

, then the layer is part of Saturn.

<span class='latex-bold'>(E) </span>

v \propto R^2

, then the layer is part of the satellites of Saturn.

1

2013 F=ma Problem 12

A spherical shell of mass

M

R

is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle

\theta

with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is

g

. The moment of inertia of a thin shell of radius

r

m

about the center of mass is

I = \frac{2}{3}mr^2

; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is

I = \frac{2}{5}mr^2

.

<span class='latex-bold'>(A) </span> g \sin \theta \\ <span class='latex-bold'>(B) </span> \frac{3}{4} g \sin \theta\\ <span class='latex-bold'>(C) </span> \frac{1}{2} g \sin \theta\\ <span class='latex-bold'>(D) </span> \frac{3}{8} g \sin \theta\\ <span class='latex-bold'>(E) </span> \frac{3}{5} g \sin \theta

1

2013 F=ma Problem 11

A right-triangular wooden block of mass

M

is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass

m

, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table?

<span class='latex-bold'>(A) </span> 2mg\\ <span class='latex-bold'>(B) </span> 2mg + Mg\\ <span class='latex-bold'>(C) </span> mg + Mg\\ <span class='latex-bold'>(D) </span> Mg + mg( \sin \alpha + \sin \beta)\\ <span class='latex-bold'>(E) </span> Mg + mg( \cos \alpha + \cos \beta)

1

2013 F=ma Problem 10

Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass?

<span class='latex-bold'>(A)</span>

Measurements of the orbit of a test mass around the object.

<span class='latex-bold'>(B)</span>

Measurements of the time it takes the object to roll down an inclined plane.

<span class='latex-bold'>(C)</span>

Measurements of the tidal forces applied by the object to a liquid body.

<span class='latex-bold'>(D)</span>

Measurements of the behavior of the object as it oats in water.

<span class='latex-bold'>(E)</span>

Measurements of the force applied to the object by a uniform gravitational field.

1

2013 F=ma Problem 9

A truck is initially moving at velocity

v

. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force

F

to the truck. The truck rolls a distance

x

before coming to a stop, and the time it takes to stop is

t

.
Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)?

<span class='latex-bold'>(A) </span> Fx\\ <span class='latex-bold'>(B) </span> Ft/2\\ <span class='latex-bold'>(C) </span> Fxt\\ <span class='latex-bold'>(D) </span> 2Ft\\ <span class='latex-bold'>(E) </span> 2Fx/v

1

2013 F=ma Problem 8

A truck is initially moving at velocity

v

. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force

F

to the truck. The truck rolls a distance

x

before coming to a stop, and the time it takes to stop is

t

.
Which of the following expressions is equal the initial kinetic energy of the truck (i.e. the kinetic energy before the driver starts braking)?

<span class='latex-bold'>(A) </span> Fx\\ <span class='latex-bold'>(B) </span> Fvt\\ <span class='latex-bold'>(C) </span> Fxt\\ <span class='latex-bold'>(D) </span> Ft\\ <span class='latex-bold'>(E) </span> \text{Both (a) and (b) are correct}

1

2013 F=ma Problem 7

A light car and a heavy truck have the same momentum. The truck weighs ten times as much as the car. How do their kinetic energies compare?

<span class='latex-bold'>(A)</span>

The truck's kinetic energy is larger by a factor of

100

<span class='latex-bold'>(B)</span>

They truck's kinetic energy is larger by a factor of

10

<span class='latex-bold'>(C)</span>

They have the same kinetic energy

<span class='latex-bold'>(D)</span>

The car's kinetic energy is larger by a factor of

10

<span class='latex-bold'>(E)</span>

The car's kinetic energy is larger by a factor of

100

1

2013 F=ma Problem 6

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.
How tall is the building?

<span class='latex-bold'>(A) </span> 50 \text{ m}\\ <span class='latex-bold'>(B) </span> 80 \text{ m}\\ <span class='latex-bold'>(C) </span> 100 \text{ m}\\ <span class='latex-bold'>(D) </span> 150 \text{ m}\\ <span class='latex-bold'>(E) </span> 400 \text{ m}

1

2013 F=ma Problem 5

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.
At what time(s) does the student have maximum downward velocity?

<span class='latex-bold'>(A)</span>

At all times between

2 s

4 s

<span class='latex-bold'>(B)</span>

4 s

<span class='latex-bold'>(C)</span>

At all times between

4 s

22 s

<span class='latex-bold'>(D)</span>

22 s

<span class='latex-bold'>(E)</span>

At all times between

22 s

24 s

1

2013 F=ma Problem 4

The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is

\mu

. Which of the following gives the maximum value of

\theta

such that the sign will not collapse?

<span class='latex-bold'>(A) </span> \sin \theta = 2 \mu \\ <span class='latex-bold'>(B) </span> \sin \theta /2 = \mu / 2\\ <span class='latex-bold'>(C) </span> \tan \theta / 2 = \mu\\ <span class='latex-bold'>(D) </span> \tan \theta = 2 \mu \\ <span class='latex-bold'>(E) </span> \tan \theta / 2 = 2 \mu

1

2013 F=ma Problem 3

Tom throws a football to Wes, who is a distance

l

away. Tom can control the time of flight

t

of the ball by choosing any speed up to

v_{\text{max}}

and any launch angle between

0^\circ

90^\circ

. Ignore air resistance and assume Tom and Wes are at the same height. Which of the following statements is incorrect?

<span class='latex-bold'>(A)</span>

v_{\text{max}} < \sqrt{gl}

, the ball cannot reach Wes at all.

\\

<span class='latex-bold'>(B)</span>

Assuming the ball can reach Wes, as

v_{\text{max}}

l

held fixed, the minimum value of

t

\\

<span class='latex-bold'>(C)</span>

Assuming the ball can reach Wes, as

v_{\text{max}}

l

held fixed, the maximum value of

t

\\

<span class='latex-bold'>(D)</span>

Assuming the ball can reach Wes, as

l

v_{\text{max}}

held fixed, the minimum value of

t

\\

<span class='latex-bold'>(E)</span>

Assuming the ball can reach Wes, as

l

v_{\text{max}}

held fixed, the maximum value of

t

1

2013 F=ma Problem 2

Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall.

<span class='latex-bold'>(A) </span> 11 \text{ m/s}\\ <span class='latex-bold'>(B) </span> 15 \text{ m/s}\\ <span class='latex-bold'>(C) </span> 19 \text{ m/s}\\ <span class='latex-bold'>(D) </span> 30 \text{ m/s}\\ <span class='latex-bold'>(E) </span> 35 \text{ m/s}

1

2013 F=ma Problem 1

An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes

5

seconds for the first car to pass the observer. How long will it take for the

10\text{th}

<span class='latex-bold'>(A)</span> \hspace{1mm} 1.07s\\ <span class='latex-bold'>(B)</span> \hspace{1mm } 0.98s\\ <span class='latex-bold'>(C)</span>\hspace{1mm} 0.91s\\ <span class='latex-bold'>(D)</span>\hspace{1mm} 0.86s\\ <span class='latex-bold'>(E)</span>\hspace{1mm} 0.81s