Advertisements
Advertisements
Question
A particle starts from the origin at t = 0 s with a velocity of 10.0 `hatj "m/s"` and moves in the x-y plane with a constant acceleration of `(8.0 hati + 2.0 hatj) ms^(-2)`.
- At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
- What is the speed of the particle at the time?
Advertisements
Solution 1
Velocity of the particle `vecv = 10.0 hatj` m/s
Acceleration of the particle = `veca = (8.0 hati + 2.0 hatj)`
Also
But `veca = (dvecv)/(dt) = 8.0 hati +2.0 hatj`
`(dvecv) = (8.0 hati + 2.0 hatj)dt`
Integrating both sides:
`vecv(t)= 8.0t hati + 2.0t hatj + vecu`
where
`vecu` = velocity vector of the particle at t= 0
`vecv` = velocity vector of the particle at time t
But `vecv = (dvecr)/(dt)`
`dvecr = vecvdt = (8.0t hati + 2.0t hatj + vecu)dt`
Integrating the equations with the conditions: at t = 0; r = 0 andat t = t; r = r
`vecr = vecut + 1/28.0t^2 hati + 1/2xx2.0t^2 hatj`
`=vecut + 4.0t^2 hati + t^2 hatj`
`=(10.0 hatj)t + 4.0t^2 hati + t^2 hatj`
`x hati + y hatj = 4.0t^2 hati + (10t + t^2)hatj`
Since the motion of the particle is confined to the x-y plane, on equating the coefficients of `hati "and" hatj`, we get:
`x = 4t^2`
`t = (x/4)^(1/2)`
And `y = 10t + t^2`
(a) When x = 16 m
`t=(16/4)^(1/2)= 2s`
∴ y = 10 × 2 + (2)2 = 24 m
(b) Velocity of the particle is given by:
`vecv(t) = 8.0t hati + 2.0t hatj + hatu`
at t = 2s
`vecv(t) = 8.0 xx 2 hati + 2.0 xx 2 hatj + 10 hatj`
=`16 hati+ 14 hatj`
∴Speed of the particle
`|vecv| = sqrt((16)^2 + (14)^2)`
`=sqrt(256+196) = sqrt(452)`
`= 21.26 "m/s"`
Solution 2
it is given that `vecr_(t = 0s) = vecv_(0) = 10.0 hatj` m/s and `veca(t) = (8.0 hati + 2.0 hatj) ms^(-2)`
(a) it means `x_0 = 0,u_x = 0, a_x = 8.0` `ms^(-2)` and x = 16 m
Using relation `s = x - x_0 = u_xt+1/2a_xt^2` we have
`16 - 0 = 0 + 1/2 xx 8.0 xx t^2 => t = 2s`
`:.y = y_0 + u_yt+ 1/2a_yt^2 = 0 + 10.0xx2+1/2xx2.0xx(2)^2`
= 20 + 4 = 24 m
(b) Velocity of particle at t= 2 s along x-axis
`v_x = u_x+a_xt=0 + 8.0 xx 2 = 16.0` m/s
and along y-axis `v_y = u_y+a_yt = 10.0 + 2.0 xx 2 = 14.0` m/s
∴Speed of particle at t = 2s
`v= sqrt(v_x^2+v_y^2) = sqrt((16.0)^2+(14.0)^2) = 21.26 ms^(-1)`
APPEARS IN
RELATED QUESTIONS
A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone?
A piece of stone tied at the end of a thread is whirled in a horizontal circle with uniform speed by hand. Answer the following questions:
- Is the velocity of stone uniform or variable?
- Is the acceleration of stone uniform or variable?
- What is the direction of acceleration of stone at any instant?
- Which force provides the centripetal force required for circular motion?
- Name the force and its direction which acts on the hand.
Which of the following quantity remains constant in uniform circular motion:
A small sphere is attached to a cord and rotates in a vertical circle about a point O. If the average speed of the sphere is increased, the cord is most likely to break at the orientation when the mass is at ____________.
A particle performs uniform circular motion in a horizontal plane. The radius of the circle is 8 cm. The centripetal force acting on the particle is 15 N. Its kinetic energy is ____________.
A particle is moving in uniform circular motion with speed 'V' and radius 'R'. The angular acceleration of the particle is ______.
At any instant, the magnitude of the centripetal force on a particle of mass 'm' performing circular motion is given by (ω = angular velocity and v = linear velocity of the particle) ______.
A particle is perfonning uniform circular motion. If 'θ', 'ω', 'α' and ' a' are its angular displacement, angular velocity, angular acceleration, and centripetal acceleration respectively, then which of the following is 'WRONG'?
A mass 'm' is tied to one end of a spring and whirled in a horizontal circle with constant angular velocity. The elongation in the spring is 1 cm. If the angular speed is doubled, the elongation in the spring is 6 cm. The original length of the spring is ______.
A stone of mass 3 kg attached at one end of a 2m long string is whirled in horizontal circle. The string makes an angle of 45° with the vertical then the centripetal force acting on the string is ______.
(g = 10 m/s2 , tan 45° = 1)
Statement A: Uniform circular motion is a case of accelerated motion
Statement B: In the third equation of motion we do not have the term time
If a body is moving in a circle of radius r with a constant speed v, its angular velocity is ______.
For a particle performing uniform circular motion, choose the correct statement(s) from the following:
- Magnitude of particle velocity (speed) remains constant.
- Particle velocity remains directed perpendicular to radius vector.
- Direction of acceleration keeps changing as particle moves.
- Angular momentum is constant in magnitude but direction keeps changing.
A particle is given an initial speed u inside a smooth spherical shell of radius R = 1 m that it is just able to complete the circle. Acceleration of the particle when its velocity is vertical is ______.
A ceiling fan rotates about its own axis with some angular velocity. When the fan is switched off the angular velocity becomes `(1/4)^"th"` of the original in time 't' and 'n' revolutions are made in that time. The number of revolutions made by the fan during the time interval between switch off and rest are ______. (Angular retardation is uniform)
A particle is performing a uniform circular motion along a circle of radius R. In half the period of revolution, its displacement and distance covered are respectively.
Two bodies of masses 10 kg and 5 kg moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal accelerations is ______.
The acceleration of a point on the rim of a flywheel 1 m in diameter, if it makes 1200 rpm is ______.
In uniform circular motion, although the speed is constant, why does acceleration occur?
What is the direction of the velocity of an object at any point during uniform circular motion?
