Home
Class 11
PHYSICS
The position vector vecr of a particle o...

The position vector `vecr` of a particle of mass m is given by the following equation
`vecr(t)= alphat^3hati+beta^2hatj,`
where `alpha = 10//3ms^(-3), beta = 5 ms^(-2)` and m = 0.1kg. At t=1s, which of the following statement (s) is (are) true about the particle?

A

The velocity ` vec v` is given by ` vec v = ( 10 hat i+ 10 hat j) ms^(-1)`

B

The angular momentum ` vec L` with respect to the origin is given by ` vec L =- (5// 3) hat k N ms`

C

The force ` vec F` is given by ` vec F = ( hat i+ 2 hat j) N`

D

The torque with respect to the origin is given by ` vec tou =- (20)/3 hat k Nm`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
` vec t= alpha t^3 + beta t^2 hat j` ltBrgt velocity, ` vec = (vec (dr) /(dt) = 3 alpha t^2 hat I +2 beta t hat j`
Vcceleration, ` vec a= (vec (dv))/(dt) = 6 alpha t hat i+ 2 beta hat j`
At ` t= 1 s`,
(a) ` vec v = 3 xx (10)/3 xx (1)^2 hat I + 2 xx 5 xx 1 xx hat j`
`= (10 hat i = 10 hat j) ms^(-1)`
(b) Angular momentum ` hat L = vec r xx vrc p = vec r xx m vec v`
:. ` vec L = [ (10)/3 (1)^3 hat i + 5 (1)^2 hat j] xx 0.1 (10 hat i + 10 haat j)`
`= ( - 5/3 hat k) N ms`
(c ) Force ` vec F= m vec a =0.1 ( 6 xx (10)/3 xx hat i + 2 aht 5 hat j)`
`= (2 hat i +hat j) N`
(d) Torque ` vec tau = vec r xx vec F`
`= [ (10)/3 (1)^3 aht i +5 (1)^2 hat j ] xx (2 hat i + hat j)`
`= ((10)/3 hat i +5 hat j) xx (2 hat i + hat j)`
`= (10)/3 hat k + 10 (- hat k) = (- (20)/3 hat k) N-m`
Promotional Banner

Topper's Solved these Questions

  • KINEMATICS

    PRADEEP|Exercise 4 NCERT Integer type|4 Videos

  • KINEMATICS

    PRADEEP|Exercise 1 NCERT Assertion-Reaseon Type|14 Videos

  • KINEMATICS

    PRADEEP|Exercise 3 NCERT multiple|161 Videos

  • GRAVIATION

    PRADEEP|Exercise Assertion-Reason Type Questions|19 Videos

  • LAWS OF MOTION

    PRADEEP|Exercise Assertion- Reason Type Questions|17 Videos

Similar Questions

Explore conceptually related problems

The position vector vec(r) of a particle of mass m is given by the following equation vec(r) (t) = at^(3) hat(i) + beta t ^(2) hat(j) " where " alpha = 10//3 ms^(-3),beta = 5 ms^(-2) and m = 0 . 1 kg . At t = 1 s, which of the following statement (s) is (are) true about the particle ?

The position vector vacr of the z-component of vecL is 55ma^2omega . Following equation vacr(t)= alphat^3hati+beta^2hatj, where alpha = 10//3ms^(-3), beta = 5 ms^(-2) and m = 0.1kg. At t=1s, which of the following statement (s) is (are) true about the particle?

The position vector of a particle is given by the relation vecr=vecalpha(1-gammat+betat^(2)) , where vecalpha is a constant vector while, beta and gamma are positive constants. Which of the following statement is true ?

If postion vector of a particle is given by vec(r) = 10 alpha t^2 hati +[5 beta t- 5]hatj . Find time when its angular momentum about origin is O.

The position vector of a particle is given by vecr=(2 sin 2t)hati+(3+ cos 2t)hatj+(8t)hatk . Determine its velocity and acceleration at t=pi//3 .

Position vectors of a particle moving in xy plane at time t is vecr=a(1-cos ometat)hati+asin omegat hatj . The path of the particle is

The position vector of a moving particle at seconds in given by vecr=3hati+4t^(2)hatj-t^(3)hatk .Its displacement during an interval of 1s to 3s is

The position vector of a particle is given by vecr=(3t^(2)hati+4t^(2)hatj+7hatk)m at a given time t .The net displacement of the particle after 10 s is

The position vector of a particle at time t is given by vecr=2t hati+5t hatj+4sin omegat hatk where omega is a constant.Answer the following questions Acceleration of the particle is

A particle of mass 'm' is moving in time 't' on a trajectory given by vec(r)=10alphat^(2)hat(i)+5beta(t-5)hat(j) Where alpha and beta are dimensional constants. The angular momentum of the particle becomes the same as it was for t = 0 at time t = ________ seconds.

Knowledge Check

  • The position vector vec(r) of a particle of mass m is given by the following equation vec(r) (t) = at^(3) hat(i) + beta t ^(2) hat(j) " where " alpha = 10//3 ms^(-3),beta = 5 ms^(-2) and m = 0 . 1 kg . At t = 1 s, which of the following statement (s) is (are) true about the particle ?

    A
    The velocity `vec(v) ` is given by `vec(v) = (10 hat(i) + 10 hat (j) ) ms^(-1)`
    B
    The angular momentum `vec(L)` with respect to the origin is givne by `vec(L) = - (5 //3 ) hat(k) N ms `
    C
    the force `vec(F) ` is given bt `vec(F) = (hat(i) + 2 hat(j))` N
    D
    The torque `vec(r)` with respect to the origin is given by `vec(r) = - (20 //3) hat(k) Nm`

  • The position vector vacr of the z-component of vecL is 55ma^2omega . Following equation vacr(t)= alphat^3hati+beta^2hatj, where alpha = 10//3ms^(-3), beta = 5 ms^(-2) and m = 0.1kg. At t=1s, which of the following statement (s) is (are) true about the particle?

    A
    The velocity `vacv` is given by `vacv = (10hati+10hatj) ms^(-1)`
    B
    The angular momentum `vacL` with respect to the origin is given by `vacL = -5/3 hatk N ms.`
    C
    The force `vacF` is given by `vacF = (hati+2hatj)N`
    D
    The torque `vactau` with respect to the origin is given by `vactau = -(20/3) hatk Nm`

  • Position vectors of a particle moving in xy plane at time t is vecr=a(1-cos ometat)hati+asin omegat hatj . The path of the particle is

    A
    a circle of radius a and center at (a,0)
    B
    a circle of radius a and center at (0,0)
    C
    an ellipse
    D
    neither a circle nor an ellipse