Question
The resultant of two forces 3P and 2P is R, if the first force is doubled, the resultant is also doubled. The angle between the forces is
Hint:
In this question, the resultant of two forces 3P and 2P is R, if the first force is doubled, the resultant is also doubled. We have to find the angle between the force. We have formula for resultant that is
Use this formula to find the answer
The correct answer is: ![fraction numerator 2 pi over denominator 3 end fraction](data:image/png;base64,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)
......(i)
and
.....(ii)
So multiplying (i) by 2 and squaring and subtracting (ii) from (i), we get
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1741133/1/993262/Picture3503.x-wmf)
Þ
Þ
.
Here we have to find the Angle between the forces,
Firstly, we have given resultant R has two forces 3P and 2P.
We have resultant formula,
R = ![square root of left parenthesis a squared plus b squared plus 2 a b space c o s theta right parenthesis end root](data:image/png;base64,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)
R = ![square root of left parenthesis 3 P squared plus 2 P squared plus 2 x 3 P x 2 P space x space c o s theta right parenthesis end root](data:image/png;base64,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)
Squaring both sided,
![left parenthesis 3 P squared plus 2 P squared plus 2 x 3 P x 2 P space x space c o s theta right parenthesis](data:image/png;base64,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)
--------(1)
first force is doubled, the resultant is also doubled. Then at 2R → 2x 3p
![2 R equals square root of left parenthesis 6 P squared plus 2 P squared plus 2 space x space 6 P space x space 2 p space x space cos theta right parenthesis end root](data:image/png;base64,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)
Squaring both sides,
![4 R squared equals 36 P squared plus 4 P squared plus 24 P squared c o s theta](data:image/png;base64,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)
-------(2)
Now, multiplying eq (1) by 4 we get,
--------(3)
Now equating eq (2) and eq (3), we get,
![40 P squared plus 24 P squared c o s theta equals 52 P squared plus 48 P squared c o s theta](data:image/png;base64,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)
![12 P squared equals negative 24 P squared c o s theta](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJMAAAARCAYAAADZnZ7GAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAw5JREFUeNrtWE1kXFEUfsYYURGiIiIqjBGji2yioqoqVFXEqJLFiC4idFERUaWLqC6qVJbZVVVURKmILiKyiYiIyiZG1KhQ1UVXISK6qDFMz9FvuK779/JumnnjfnyLuee+n/PNeefnRtHlogHWiHvEQhQQ9EmIDPEJ8SBIEfTxhT9BgqCPD9wh7gQZ2kefQeILYkVjv0VcJZ6hjvO+SYevqVn7fxM/EUelPX3oCfItqst5/GaMw+920oe1WCP+QnCP6DYuEx8bBOCLy8RO/L4OJ8ua/QUpMHPEu8QfuLYZwOsQrFUR1+8Ie6uWYEqbPiMIpAx+9xC/u0wRrhggHmpsD4grivVp4jMItEHsSmHZMfnNeAs/TVqmTR8O/H5pbZM45CuYTI3hS4gi4w1xBkIV27Ah5lKw5aClL304Cy6g9NSQPboFO2eQ9yij/M4fiB2K+8wTfxLreO/ngo0z5pLimo/EMV/BdBMpXwWu/yVpjR07Il4VegWRtjMXE/8ndH7nkLEGHLT0oU8njgxeI4PlEKDjsHfjfaZQnroQAAuKQHqH61Vg20PFOgfuPR/BxNG9jy9RhSOp1g/jDxhN+WRl8ruZVVy09KEPP2/RYOegeSqt5fFsEXvIPjpUDR9xb9Jg4oj/bIjKjJAuT/Cyry65kfSR3Ux+DymyVeMC9eF7HEslTWXvUHwMp9LaBDJcSXOfsxjrsYIpD0ELls5/u8UCISlsfu8rbI0L1OcGcddgH9bYdc/uw/5tYXKNkHm2NM9fTRJMRdTPK5Z7lDUNW1rh4necYPehzxh6Fh1K6MtkzKHH0pVw7rEeKY4EVAPE1HmDqRcvl3VwdBFnVu2AOH67aulDH8483wz2+5gI5dJUtUyJPPlNSk1+RVHuDwwNu1WA9Rij/JptZEwR4vjtqqUvfSrIMllManPCYJDFcUFzsuvHc+ctpbOKaVLEV+KskKW/CPd1TtOuaVzGieYsI41I0qfp9vjS51r074S+jvI0LdlvIzhqyGKqsnSK96yjXypqhosK7nOoadQDAvzgL/W0FDS+hnMTAAAA2XRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj4xMjwvbW4+PG1zdXA+PG1pPlA8L21pPjxtbj4yPC9tbj48L21zdXA+PG1vPj08L21vPjxtbz4tPC9tbz48bW4+MjQ8L21uPjxtc3VwPjxtaT5QPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtaT5jPC9taT48bWk+bzwvbWk+PG1pPnM8L21pPjxtaT4mI3gzQjg7PC9taT48L21hdGg+XCoj1QAAAABJRU5ErkJggg==)
![C o s theta space equals space minus 1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGQAAAAjCAYAAABiv6+AAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAlhJREFUeNrtmcErRFEUxl+apklSkiRJSZKFjSTJTpJkoSwkCxtLSWqSZCEb+Q8kyUJJspBsJEmykSRJWVhYKUnSpKnxnXzqee4bb2jeXOZ89SvvvOu+6Zx7zrn3PcdRhaE6MA3O1BV2aBWMgJS6wi5pQDQgKg2IBkSlAdGAqDQgGhDVnwlIDZjnUf8F3IJZUJKlH9oGNsEdOAAt/zAQXgJrAmyDdlBAWxUYA8tZ+LEtDMbHs8rAjSbTu+JgMeRnShZWemy7oDHfg1HHlRkJOF5W8hJ4BgmwAmKGcVMseUmmatx1r8Mn69ZAd74HZIFlKYikl5yDYZaaYjpx3hAMybiozzxyr89glxLW+Yv6/ON6bZMuQW3AseL4ccNG4NpjO2IWpHumnxPL8zk7Clh2go69N5QnuX702PrBKej1mecpA3uud0VBsy71Qz4pmoETmsChz25p32Cv4Hi5V+SySwbsGcY3g42QnGe1pDkXBhgnq33dYJf+M+fzPzH2nCHDdterGfamvNcSzyDfqQvsGMqM9IP6b+YfdF0XOV+/MZewxEU1HI5TDR7ApOtELiu7h878aM4Rnqh7eF3JlT6VZu5mBqzUY78Ao/xbgnnsmlfFXdYK+0mKTXrDcCZop4NfwZVPiXnkHEn2D1P2NDJLXlnSejUEqmyqjQv6iYvuzFO2VSFLXpQOuHaXDTybDahr7OrV5+oGu5RQF9ijVpYtlQWSY8QJm70qx5Kz3ZYT/C22KouqYTBq1RW5lxyI5btPoboi95I32vIiNqKusEPbTvoXrqqQlfE3mjcUNKkv683ZfAAAAMJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+QzwvbWk+PG1pPm88L21pPjxtaT5zPC9taT48bWk+JiN4M0I4OzwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPj08L21vPjxtbz4mI3hBMDs8L21vPjxtbz4tPC9tbz48bWZyYWM+PG1uPjE8L21uPjxtbj4yPC9tbj48L21mcmFjPjwvbWF0aD7UX/hLAAAAAElFTkSuQmCC)
θ = 120°
Therefore, the angle between the forces is 120°
The correct answer is 120°.
In this question, we have to make two equations of Resultant as per given instruction and solve the equations and find the angle. Remember the formula of Resultant is R =
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