Let us present another series of problems from the open FIPI bank with physical content, in solving which you need to be able to count and know a little trigonometric functions.
Task No. 28675 A load weighing 0.16 kg oscillates on a spring with a speed varying according to law (1), where t- time since the start of oscillations, T=2 s - oscillation period, v 0 =1.5 m/s. Kinetic energy E(in joules) of the load is calculated using formula (2), where m- mass of cargo in kilograms, v— speed of the load (in m/s). Find the kinetic energy of the mass 3 seconds after the start of oscillation. Give your answer in joules.
Task No. 28687 Load weighing 0.4 kg oscillates on a spring with a speed varying according to the lawv( t)=0.5cosπt, Where tm— cargo mass (in kg),v— speed of the load (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 0.025 J. Express your answer decimal, if necessary, round to the nearest hundredth.
Task No. 28697 The speed of a load oscillating on a spring varies according to the law v(t)=7sin( πt/ 4) (cm/s), where t— time in seconds. What percentage of the first two seconds was the speed greater than 3.5 cm/s? Express your answer as a decimal fraction; if necessary, round to the nearest hundredth.
Solution.Let us find after what amount of time the speed of the oscillating load becomes equal to 3.5 cm/s.
3.5=7sin( πt/ 4), sin( πt/ 4)=0.5 or πt/ 4= π/ 6, multiplied by 12 and divided by π we get both sides of the equation 3t=2, t=0.67. The remaining time up to 2 seconds, the speed was greater, that is, 2-0.67 = 1.33.
Answer 1,33.
Tasks for independent solution.
A load weighing 0.16 kg oscillates on a spring with a speed varying according to law (1), wheretT=2 c is the oscillation period,v 0 =0,5 m/s. Kinetic energyEmv— speed of the load (in m/s). Find the kinetic energy of the load 7 seconds after the start of oscillation. Give your answer in joules.
A load weighing 0.15 kg oscillates on a spring with a speed varying according to the law (1), wheret- time since the start of oscillations,T=2 c is the oscillation period,v 0 =0,4 m/s. Kinetic energyE(in joules) of the load is calculated using formula (2), wherem- mass of cargo in kilograms,v— speed of the load (in m/s). Find the kinetic energy of the load 11 seconds after the start of oscillation. Give your answer in joules.
A load weighing 0.02 kg oscillates on a spring with a speed varying according to the law (1), wheret- time since the start of oscillations,T=2 c is the oscillation period,v 0 =1 m/s. Kinetic energyE(in joules) of the load is calculated using formula (2), wherem- mass of cargo in kilograms,v— speed of the load (in m/s). Find the kinetic energy of the mass 17 seconds after the vibration begins. Give your answer in joules.
A load weighing 0.8 kg oscillates on a spring with a speed varying according to law (1), wheret- time since the start of oscillations,T=2 c is the oscillation period,v 0 =0,5 m/s. Kinetic energyE(in joules) of the load is calculated using formula (2), wherem- mass of cargo in kilograms,v— speed of the load (in m/s). Find the kinetic energy of the mass 21 seconds after the vibration begins. Give your answer in joules.
A load weighing 0.2 kg oscillates on a spring with a speed varying according to law (1), wheret- time since the start of oscillations,T=2 c is the oscillation period,v 0 =0,6 m/s. Kinetic energyE(in joules) of the load is calculated using formula (2), wherem- mass of cargo in kilograms,v— speed of the load (in m/s). Find the kinetic energy of the mass 25 seconds after the vibration begins. Give your answer in joules.
Load weighing 0.08 v(t)=0.5cos πt, Where t— time in seconds. The kinetic energy of the load is calculated using formula (2), wherem— cargo mass (in kg),v 5 ⋅ 10 −3
Load weighing 0.16 kg oscillates on a spring with a speed varying according to the lawv(t)=1.5cos πt, Where t— time in seconds. The kinetic energy of the load is calculated using formula (2), wherem— cargo mass (in kg),v— speed of the load (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 9 ⋅ 10 −2 J. Express your answer as a decimal fraction, if necessary, round to the nearest hundredth.
Load weighing 0.15 kg oscillates on a spring with a speed varying according to the lawv(t)=0.4cos πt, Where t— time in seconds. The kinetic energy of the load is calculated using formula (2), wherem— cargo mass (in kg),v— speed of the load (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 3 ⋅ 10 −3 J. Express your answer as a decimal fraction, if necessary, round to the nearest hundredth.
Load weighing 0.2 kg oscillates on a spring with a speed varying according to the lawv(t)=0.6cos πt, Where t— time in seconds. The kinetic energy of the load is calculated using formula (2), wherem— cargo mass (in kg),v— speed of the load (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 9 ⋅ 10 −3 J. Express your answer as a decimal fraction, if necessary, round to the nearest hundredth.
v(t)= 5 sin πt(cm/s), where t— time in seconds. What fraction of the first second did the speed exceed 2,5 cm/s? Express your answer as a decimal fraction; if necessary, round to the nearest hundredth.
The speed of a load oscillating on a spring varies according to the lawv(t)= 3 sin( πt/ 4) (cm/s), where t 1,5
)= 11 sin( πt/ 5) (cm/s), where t— time in seconds. What fraction of the first second did the speed exceed 5.5 5) (cm/s), where t— time in seconds. What proportion of the first two seconds did the speed exceed 7,5 cm/s? Express your answer as a decimal fraction; if necessary, round to the nearest hundredth.The speed of a load oscillating on a spring varies according to the lawv(t)= 10 sin( πt/ 5) (cm/s), where t— time in seconds. What proportion of the first three seconds did the speed exceed 5 cm/s? Express your answer as a decimal fraction; if necessary, round to the nearest hundredth.
Date: 02-02-2015 2470Category: Physical tasks
Tag: Tasks 10
28013. A load weighing 0.08 kg oscillates on a spring with a speed varying according to the law
The kinetic energy of the load is calculated by the formula:
Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 5∙10 –3 J. Express the answer as a decimal fraction, if necessary, round to hundredths.
< t < 1, следовательно 0 < Пt < П (умножаем все части неравенства на Пи). Отметим, что на этом интервале имеет как положительное, так и negative meaning. Next, we determine what period of time in the first second the kinetic energy of the load will be at least 5∙10
–3 J, that is:
Substituting v, we get:
We get two inequalities:
Let us represent the solutions to the inequalities graphically:
We do not take into account the periodicity of the cosine, since we consider the angle in the interval from 0 to Pi.
We divide the parts of the inequalities by Pi:
Thus, the kinetic energy of the load will be at least 5∙10 –3 J from the very beginning of the movement to 0.25 seconds, and from 0.75 to the end of the first second. Total time 0.25 + 0.25 = 0.5 seconds.
Answer: 0.5
28012. A load weighing 0.08 kg oscillates on a spring with a speed varying according to the law:
The kinetic energy of the load, measured in joules, is calculated by the formula:
Where m- mass of the load (in kg), v - speed of the load (in m/s). Determine what fraction of the time from the first second after the start of movement the kinetic energy of the load will be at least 5∙10 –3 J. Express the answer as a decimal fraction, if necessary, round to hundredths.
Let us pay attention to the fact that we consider the process during the first second, that is, 0< t < 1, следовательно 0 < Пt < П, (умножаем все части неравенства на Пи). Можно сделать вывод, что sin (Пt) имеет positive value. Next, we determine what period of time in the first second the kinetic energy of the load will be at least 5∙10 –3 J, that is:
Substituting v, we get:
Section Physics -> Conservation LawsTest based on Unified State Exam materials “Work and power. Conservation laws in mechanics"
1. A man grabbed the end of a homogeneous rod of mass 100 kg lying on the ground and raised this end to a height of 1 m. What work did he do?
2. A mass of mass m = 0.2 kg is tied to a thread of length l= 1 m. The thread with the load was taken away from the vertical at an angle of 60 o (see figure) and released without an initial speed. What is the kinetic energy of the load as it passes through the equilibrium position?
1) 0.5 J
3. A ball moving on a smooth horizontal surface collides with a heavier ball of the same size and mass m lying motionless on the same surface. As a result of the partially inelastic impact, the first ball stopped, and 75% of the initial kinetic energy of the first ball was converted into internal energy. What is the mass of the first ball?
4. Two carts are moving towards each other at the same speeds v. The masses of the carts are m and 2m. What will be the speed of the carts after their absolutely inelastic collision?
5. A ball on a long, light, inextensible thread oscillates, rising above the equilibrium position to a maximum height of 20 cm. The maximum kinetic energy of the ball during the oscillation process is 1 J. The mass of the ball is: