Finding Tension in a String and Speed
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1. Perform the following operations and write your answers with the appropriate number of significant figures: (a) 589.01−89.020 (b) 1.0×9.00 (c) 180/π
(d) 180.0/π (e) 3.901×10−5+1.0×10−7
2. Perform the following unit conversions, and express the answer in scientific notation: (a) 0.510 kg to g (b) 92 mi to km (c) 0.50m3 to cm3 (d) 15μs to ms (e) 60.0 km/h to m/s
3. On her way to school, a child discovered that her loonie is missing and there is a hole in her pocket. In searching for the missing coin, the child turned back and walked for 9.5 m, along the sidewalk. Then she turned forward and walked for 3.2 m and then turned back again and walked for 17.8 m before finding her loonie.
1. What is the total displacement of the child?
2. If the child spent 8.5 minutes searching for the coin, what was her average velocity?
3. What was her average speed?
4. A car was moving on the highway at a speed of 90 km/h. The driver then stepped on the gas and started to accelerate uniformly at a rate of 0.26 m/s2, and he was able to overtake a truck in 35 s. What was the final speed of the car after this time?
5. An electron in a cathode ray tube of a TV set enters a region where it accelerates uniformly from a speed of 3.5 × 104 m/s to a speed of 1.4 × 106 m/s in a distance of 2.0 cm.
1. What is the acceleration of the electron in this region?
2. How long is the electron in the region where it accelerates?
6. A hockey player is standing on his skates on a frozen pond when an opposing player skates by with the puck, moving with a constant speed of 12 m/s. After 3.0 s, the first player makes up his mind to chase his opponent and starts accelerating uniformly at 3.8 m/s2.
1. How long does it take him to catch his opponent?
2. How far does he travel before he catches up with his opponent?
7. A man standing on the sidewalk throws a set of keys vertically upward to his wife in the window of their apartment 4.0 m above. The wife catches the keys 1.5 s later.
a. With what initial speed were the keys thrown?
b. What is the velocity of the keys just before they were caught?
c. Draw a diagram showing the trajectory of the keys.
8. A ball is thrown vertically upward with an initial speed of 11 m/s. One second later, a stone is thrown vertically upward with an initial speed of 25 m/s.
a. Find the time it takes the stone to catch up with the ball.
b. Find the velocities of the stone and the ball when they are at the same height.
9. A ship sailed for 2.8 km north, then 3.5 km west, and finally 10.1 km southwest.
a. Sketch a vector diagram representing this motion.
b. What is the net displacement of the ship?
10. Three vectors, A, B and C each have a magnitude of 74 units. Their directions relative to the positive direction of the x-axis are 15°, 115° and 215°, respectively. Calculate the magnitude and direction of the vectors
a. A + B + C
b. A + B − C
c. C − 2A
11. A stone is thrown from the top of a building upward at an angle of 30° to the horizontal with an initial speed of 24 m/s. The height of the building is 48 m.
a. How long is the stone in flight?
b. What is the magnitude and direction of the stone's velocity just before it hits the ground?
12. A ski jumper travels down in a slope and leaves the ski track, moving horizontally with a speed of 25 m/s. The landing incline below her falls off with a slope of 33°.
a. How long is the ski jumper airborne?
b. Where does the ski jumper land on the incline?
13. A force F applied to a mass m1 produces an acceleration of 3.0 m/s2. The same force applied to a second object of mass m2 produces an acceleration of 1.0 m/s2.
a. Find the value of the ratio m1 / m2.
b. If m1and m2 are combined, what will be the acceleration of the combined mass (m1+ m2) under the action of the force F?
14. Three blocks are in contact with each other on a frictionless horizontal surface as shown. Force----
The masses of the blocks are m1 = 1.0 kg, m2 = 2.0 kg, and m3 = 3.0 kg. A horizontal force F = 24 N is applied to m1.
a. Find the acceleration of the three blocks.
b. Find the net force on each block.
c. Find the magnitudes of the contact forces between the blocks.
15. A 4.1-kg box is pushed along a horizontal floor by a force of magnitude 21 N at an angle θ = 35° with the horizontal. If the coefficient of kinetic friction between the block and the floor is 0.20, calculate the acceleration of the box. Be careful when calculating the normal force.
16. A 4.0-kg toboggan rests on a frictionless icy surface, and a 2.0 kg block rests on top of the toboggan. The coefficient of static friction µs between the block and the surface of the toboggan is 0.60, whereas the kinetic friction coefficient is 0.51. The block is pulled by a 30 N-horizontal force as shown. What are the magnitudes and directions of the resulting accelerations of the block and the toboggan?
30N--
17. A 100 g bead is free to slide along an 80 cm long piece of string ABC. The ends of the string are attached to a vertical pole at A and C, which are 40 cm apart. When the pole is rotated about its axis, AB becomes horizontal.
a. Find the tension in the string.
b. Find the speed of the bead at B.
18. A crate of eggs is located in the middle of the flatbed of a pickup truck. The truck is negotiating a curve in the road that may be considered as an arc of a circle of radius 35 m. If the coefficient of static friction between the flatbed and the crate is 0.66, with what maximum speed can the truck negotiate the curve without the crate sliding out during cornering?
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The expert finds the tension in a string and speed.
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For simple discussion of significant digits you can see
http://www.fordhamprep.org/gcurran/sho/sho/lessons/lesson23.htm
1.
Perform the following operations and write your answers with the appropriate number of significant figures: (a) 589.01−89.020 (b) 1.0×9.00 (c) 180/π
(d) 180.0/π (e) 3.901×10−5+1.0×10−7
The concept of significant digits is that in which we want to maintain the order of magnitude (OM) of the absolute error in case of addition/subtraction or relative error in case of multiplication/division error in our result.
The absolute error in a number is of its last digit
The error for example in the number 4553.224 is 0.001.
Let's look at (a).
Here we have two numbers, 589.01 and 89.020. This is addition, so we need to maintain absolute error magnitude.
The absolute error's OM in 589.09 is
The absolute error's OM of 89.020 is
We cannot have the result more accurate than the least accurate factors. Thus, the resultant absolute error OM is , so we must maintain the numbers up to two digits after the decimal:
Keeping the second digit after the decimal:
Note that we are interested only in the order of magnitude.
Now let's look at (b).
We have a multiplication operation - so we must look at the relative error OM.
The most significant digit in 9.00 is the "9" which has OM of (units)
The least significant digit order of magnitude is (hundredth).
Here we have two numbers. 9.00 has a relative error of
The number 1.0 has a relative error of
Then the relative error OM of the result must be (the level of accuracy of the result is the accuracy of the least accurate factor).
Hence:
(c)
The number 180 has a relative error of . We can write to whatever accuracy we want. If we want to maintain a relative error of we see that 3.14... has the order of magnitude of , so to get a relative error of we write:
So we keep in the value of the first two digits after the decimal:
And:
The magnitude of 57.3248.. is so to maintain a relative error of we write:
So we keep in the result only the first digit after the decimal:
(d)
With exponents involved, we want to have the exponents the same.
We write:
The magnitude of 3.901×10−5 is . The magnitude of the last digit is , thus the absolute error is
So the resultant's absolute error OM is
When we add them together we get:
2.
Perform the following unit conversions, and express the answer in scientific notation: (a) 0.510 kg to g (b) 92 mi to km (c) 0.50m3 to cm3 (d) 15μs to ms (e) 60.0 km/h to m/s
When we convert units what we do we actually multiply the number by "1" in such a way that the units cancel out until we get the required final unit.
For example, we know that 1kg = 1000g
So we can write:
When we want to convert 0.510 kg to grams we simply use:
Or in scientific notation:
(b)
To convert from miles to kilometers we use:
Then:
(c)
We note that:
Then:
(d)
We note that:
Then:
(e)
When we want to convert a combination of units we use consecutive multiplications:
We have:
Then:
3.
On her way to school, a child discovered that her loonie is missing and there is a hole in her pocket. In searching for the missing coin, the child turned back and walked for 9.5 m, along the sidewalk. Then she turned forward and walked for 3.2 m and then turned back again and walked for 17.8 m before finding her loonie.
1. What is the total displacement of the child?
2. If the child spent 8.5 minutes searching for the coin, what was her average velocity?
3. What was her average speed?
Displacement is the distance and direction (it is a vector quantity) between an initial point and a final point regardless of the path taken from the initial point to the final:
The displacement is the red thick arrow.
We set the initial position of the child as the origin. We also set that going backwards will have a negative sign and moving forward has a positive sign (we are taking vectors that have directionality)
Then, her total displacement is ( is a unit vector in the positive direction):
Her displacement was 24.1 meters backwards. Note that the total distance (the length of the path) she walked was:
Velocity is a vector as well. It has a magnitude and direction. It is defined as the change in displacement over change of time.
In our case:
The average speed is not a vector. It is defined as the total length of the actual path traveled divided by the total time the travel took:
General Theory
Acceleration is the rate of change of the velocity with respect to time:
Integrating both sides gives:
...
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