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The magnitudes of vectors are 3, 4 and 5 units respectively. If , thenthe angle between and is
Given,
= 3
= 4
= 5
We have a relation,
Squaring both side, we get
⇒ ( )^{2 } = ()^{2}
⇒( )^{2}+ ()^{2} + 2 Cosθ = ()^{2}
⇒ (3)^{2} + (4)^{2} + 2.3.4 Cosθ = (5)^{2}
⇒ 9 + 16 + 24 Cosθ = 25
⇒ 24 Cosθ = 25
⇒ 24Cosθ = 25  25
⇒ θ = Cos^{1} (0)
⇒ θ = 90°
i.e π/2
The angle between is θ. The value of the triple product is [1989]
Note that . Hence their dot product is zero.
A bus is moving on a straight road towards north with a uniform speed of 50 km/hour turns through 90°. If the speed remains unchanged after turning, the increase in the velocity of bus in the turning process is [1989]
= 50 km h^{1} due North;
= 50 km h^{1 }due West. Angle between
= 50 km h^{1 } due South
∴ Change in velocity
= 70.7 km/h
The direction of this change in velocity is in SouthWest.
The maximum range of a gun of horizontal terrain is 16 km. If g = 10 ms^{–2}, then muzzle velocity of a shell must be [1990]
= 16000 [16km = 16000m]
or v = (16000g )^{½ }= (16000 x10)^{½}
= 400 ms^{ –1}
Two bodies of same mass are projected with the same velocity at an angle 30° and 60° respectively. The ratio of their horizontal ranges will be[1990]
Horizontal range is same when angle of projection is θ or (90° – θ).
An electricfan has blades of length 30cm measured from the axis of rotation. If the fan is rotating at 120 rpm, the acceleration of a point on the tip of the blade is [1990]
When milk is churned, cream gets separated due to[1991]
Cream gets separated from a churned milk due to centrifugal force.
The resultant of will be equal to [1992]
When a vector is multiplied with a scalar, the result is a vector.
If a body A of mass M is thrown with velocity v at an angle of 30° to the horizontal and another body B of the same mass is thrown with the same speed at an angle of 60° to the horizontal, the ratio of horizontal range of A to B will be [1992]
Horizontal range is same wh en an gle of projection with the horizonatal is θ and (90° – θ).
A boat is sent across a river with a velocity of 8 km h^{–1}. If the resultant velocity of boat is 10 km h^{–1} , then the velocity of the river is [1993]
= 6 km h ^{–1}11.
A body constrained to move in ydirection, is subjected to a force given by . What is the workdone by this force in moving the body through a distance of 10m along yaxis ? [1994]
Since displacement is along the ydirection, hence displacement
Work done
When a body moves with a constant speed along a circle[1994]
On circular motion, the force acts along the radius and displacement at a location is perpendicular to the radius i.e., θ = 90° As work done =
Th e position vector of a par ticle is . The velocity of the particle is [1995]
Slope of position vector = & slope of velocity vector
∴ velocity is perpendicular to the displacement.
Which of the following is not a vector quantity?
A vector quantity has both magnitude and direction. In the given options, speed has only magnitude, therefore, it is non vector or scalar quantity.
A bullet is fired from a gun with a speed of 1000 m/s in order to hit a target 100 m away. At what height above the target should the gun be aimed? (The resistance of air is negligible and g = 10 m/s^{2}) [1995]
Speed of the bullet (v) = 1000 m/s and horizontal distance of the target (s) = 100 m.
Time taken to cover the horizontal distance
During this time, the bullet will fall down vertically due to gravitational acceleration.
A body is whirled in a horizontal circle of radius 20 cm. It has an angular velocity of 10 rad/s.What is its linear velocity at any point on circular path[1996]
Radius of circular path = 20 cm =
Angular speed of body = 10 rad/s
Linear velocity = radius × Angular speed
A ball whose kinetic energy is E, is thrown at an angle of 45º with the horizontal. Its kinetic energy at the highest point of its flight will be [1997]
Kinetic energy is E
Velocity at highest point
[∴ at the highest point there is only horizontal velocity]
K.E. at highest point
Find the torque of a force acting at the poin . [1997]
Two particles A an d B are con nected by a rigid rod AB. The rod slides along perpendicular rails as shown here. The velocity of A to the left is 10 m/s. What is the velocity of B when angle α = 60º? [1998]
Let after 1 sec angle become 60°. When the end A moves by 10 m left, the end B moves upward by BB' = = 17.3 m/s
A ball of mass 0.25 kg attached to the end of a string of length 1.96 m is moving in a horizontal circle. The string will break if the tension is more than 25 N. What is the maximum speed with which the ball can be moved? [1998]
A 500 kg car takes a round turn of radius 50 m with a velocity of 36 km/h. The centripetal force is[1999]
V =36km/h=10m/s
Centripetal force
= 1000 N
What is the linear velocity if angular velocity vect or and position vector [1999]
As we know that
The angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units, is[1999]
We know that, R^{2} = A^{2} + B^{2} + 2 AB cosθ
(24)^{2} = (12)^{2} + (18)^{2} + 2(12)(18) cos θ
A body of mass 0.4 kg is whi rled in a vertical circle making 2 rev/sec. If the radius of the circle is 1.2 m, then tension in the string when the body is at th e top of the circle, is [1999]
Given : Mass (m) = 0.4 kg Its frequency (n) = 2 rev/sec Radius (r) =1.2 m.
We know that linear velocity of the body (v) = ωr = (2πn)r
= 2 × 3.14 × 1.2 × 2 = 15.08 m/s.
Therefore, tension in the string when the body is at the top of the circle (T)
= 45.78  3.92 = 41.56 N
A stone tied with a string, is rotated in a vertical circle. The minimum speed with which the string has to be rotated [1999]
Minimum speed with which the string is rotating in a vertical circle (v) = The minimum speed of stone is independent of mass of stone.
A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is 0.5 m/s at an angle 120º with the direction of flow of water. The speed of water in stream is [1999]
Velocity of person
If a unit vector is represented by , the value of c is [1999]
Two projectiles ar e fir ed from t he same point with the same speed at angles of projection 60º and 30º respectively. Which one of the following is true? [2000]
Given, u_{1} = u_{2 }= u, θ_{1} = 60º, θ_{2} = 30º In 1^{st} case, we know that range
In 2^{nd} case, when θ_{2} = 30º, then
[we get same value of ranges].
A small sphere is attach ed 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 [2000]
In the case of a body describing a vertical circle,
Tension is maximum when cosθ = +1 and velocity is maximum Both conditions are satisfied at θ = 0º (i.e. at lowest point B]
A particle moves with a velocity under the in fluence of a constant force The instantaneous power applied to the particle is [2000]
= 6 × 20 – 4 × 15 –3 ×5 = 45 J/s
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