A ball is released from rest and rolls down an inclined plane, as shown in the following figure, requiring 4 s to cover a distance of 100 cm along the plane: Which one of the following is the correct value of angle θ that the plane makes with the horizontal? (g = 1000 cm/s 2)
θ = sin -1 (1/80)
CONCEPT:
- Equation of Kinematics : These are the various relations between u, v, a, t, and s for the particle moving with uniform acceleration where the notations are used as:
- Equations of motion can be written as
V = U + at
\(s =ut+\frac{1}{2}{at^{2}}\)
V 2 = U 2+ 2as
where, U = Initial velocity, V = Final velocity, g = Acceleration due to gravity, t = time, and h = height/Distance covered
u = Initial velocity of the particle at time t = 0 sec
v = Final velocity at time t sec
a = Acceleration of the particle
s = Distance traveled in time t sec
CALCULATION :
- Here ball is on an inclined plane of inclination θ. So we have two components of acceleration due to gravity.
- One component will be along the inclined plane which will accelerate the object (a = g sinθ) and the other will be perpendicular to the plane (g cosθ) which is of no use here in this case.
- As the ball is released from rest,
Initial velocity,u = 0 m/s, distance, S = 100 cm, time, t = 4 sec
Acceleration, a = g sinθ = 1000 sinθ cm/s 2
- Use the equation
\(⇒ S = ut + \frac{1}{2}a{t^2}\)
⇒ 100 = 0 + (1/2).1000 sinθ .4 2
⇒ 80 sinθ = 1
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