Dynamic

Spring 2021

1. A 500 kg parade float is towed with a trailer hitch on a horizontal surface with negligible rolling resistance. The float starts from rest at time=0. The force that the trailer hitch applies to the float, F(t), is given by the graph below. What is the speed and direction of the center of gravity of the float at time=20 sec?

5 rad/s 2 rad/s

!"#"= 500 kg

F(t)

R = 1 mR = 1 m

no rolling resistance

$

x

y

time [sec]

Force [N]

500

0 −250

10 15 20

7!""#$%&'%(

2. A highway ramp is three-quarters of a circle with a radius of 50m. A car enters the ramp at Point A with a speed of 5 m/s. What is the constant tangential acceleration, 𝑎", so that the car will reach 15 m/s just as it gets to the end of the ramp?

A

B

!" = 5 &/(

!) = 15 &/(

+, = ?

view from the top gravity into the page

3. A 300 kg roller-coaster car travels with negligible friction from State 0 to State 2. At State 0 its speed is 20 m/s and its elevation is 10 m. It then drops to an elevation of 5 m (State 1). Then rises to 20 m (State 2). What is the speed of the car at State 2?

!" = 10 ' !( = 5 '

!* = 20 '

,State 0

State 1

State 2

-" = 20 '//

-* = ?

4. A group of kids fire a toy rocket straight downward from a tree branch. At a particular time, the mass of the rocket is 0.5 kg, the exhaust flow rate is 0.1 kg/s with a velocity of 30 m/s relative to the rocket, directed upward. What is the acceleration of the rocket at that time?

R !"# = 0.5 )!

*+/# = 30 "/. "̇+ = 0.1 )!/.

1# = ?

A B

20°

$

%& = 0.7 %* = 0.6

5. Cart A has a mass of 10kg and is being pulled up a 20° incline with an acceleration of 2 m/𝑠'. Block B has a mass of 8 kg and is resting on Cart A. Block B is not slipping relative to Cart A. The static coefficient between Cart A and Block B is 𝜇) = 0.7 and the kinetic coefficient of friction is 𝜇+ = 0.6 What is the magnitude of the friction force between Cart A and Block B?

5 #$

%& = ? $

%) = 6 +/-

6. A load is constrained to move vertically by the pulley system shown. When the load is moving upward at 𝑣- = 6 m/s, what is the speed, 𝑣0, of the end of the cable?

! "# = 0.5

20 kg

k = 500 N/m

() = 0.4+ STATE 1

,) = 0 MOTOR

!20 kg

(- = 0.8+ STATE 2

,- = 0 MOTOR

"# = 0.5

7. A 20kg load is pulled from State 1 to State 2. The load is at rest at both State 1 and State 2. At State 1, the length of the spring is 𝑠1 = 0.4 𝑚, compared with its unstretched length of 𝑠5 = 0.3 𝑚. Between State 1 and State 2, the block is dragged across a surface with a coefficient of friction of 𝜇+ = 0.5 At State 2, the length of the spring is 𝑠' = 0.8 𝑚. What is the total work that the cable put on the load moving from State 1 to State 2?

8. An elevator starts from rest on the top floor of a building. It travels 20m downward to the bottom floor. First, it accelerates downward at 1.5 m/𝑠' for 10m, then it decelerates at 1.5 m/𝑠' for the final 10m, so that its speed will be zero again at the bottom. How much time did it take for the elevator to travel the full 20m?

∆" = 20 '

() = 0

(* = 0

s [m]

+ [m/",]

1.5

0.0

−1.5

2010

s

9. A golfer hits a ball out of a sand trap onto the green. The green is 1m higher than the sand trap. The initial velocity of the ball after it is hit is 8 m/s, directed 60° relative to horizontal. How far will the ball have traveled in the x-direction when it first hits the green?

!

60°

∆& = ?

1 m *+ = 8 -//

10. A 10 kg Cart C can move without rolling resistance on a horizontal surface. Initially (State 1) the cart is moving to the right at 3 m/s. Bag B is thrown at the cart with a velocity of 4 m/s directed 30° above horizontal. Having done a good job of projectile motion and relative motion calculations, the bag lands on the cart and does not bounce or slide off the cart. Just after the bag lands on the cart (State 2), the combined cart and bag are moving to the right a 1 m/s. What is the mass of the bag?

C

B

!"# = 3 '/)

*

!+#= 4 m/s30° State 1 bag is thrown

'. = 10 0*

C B

!"# = 1 '/)

* State 2 just after bag lands on cart

12. A 2 kg toy car is travelling with a constant speed of 5 m/s. The static coefficient of friction between the wheels and the ground is 𝜇) = 0.6 What is the minimum radius of curvature that the car can turn without slipping relative to the ground?

!"#$ = ?

( = 5 m/s

view from the top gravity into the page

*+ = 0.6 m = 2 kg

12. A bug is crawling radially outward on the spoke of a wheel at a constant speed of 4 cm/s relative to the spoke of the wheel. The center of the wheel is also moving to the left at 160 cm/s and accelerating at 120 cm/𝑠'. As a result, its angular speed is �̇� =4 rad/s and its angular acceleration is �̈� = 3 rad/𝑠'. At that same time the bug is 30 cm from the center of the wheel and is directly above the center of the wheel. What is the acceleration vector of the bug relative to the ground at that time?

! " = 4 &' /)

C !* = 160 cm/s .* = 120 cm/)0

30 cm

1̈ = 3 rad/)0 1̇ = 4 rad/s

B

x

y

5

13. Sand is being loaded onto a moving cart from a stationary container. Sand leaves the container downward with a mass flow rate of 30 kg/s and a speed of 6 m/s. The cart moves on a horizontal surface with negligible rolling resistance. The empty mass of the cart is 200 kg. At a particular time there is 100 kg of sand in the cart and it is moving to the right at 2 m/s. What is the magnitude and direction of the acceleration of the cart at that time?

no rolling resistance

!"#$% = 2 m/s

("#$% = ?

+"#$% = 200 -. +/#01 = 100 -.

x

y

Stationary

+̇40 = 30 -./7 !/#01 = 6 +/7 downward

.

14. Block A has a mass of 6 kg and is moving to the right at 2 m/s on a horizontal surface with a kinetic coefficient of friction of 𝜇+ = 0.8 A cable and simple pulley connect Block A to Block B which has a mass of 4 kg and moves vertically. Ignore the friction and mass of the cable and pulley. What is the magnitude and direction of the acceleration of Block A?

A

B

!" = 0.8 '( = 2 m/s +

,( = ?.( = 6 0+.1 = 4 0+

!