A body slides down from the rest from the top of6.4m long rough inclined plane at 30 degree with the horizontal.find the time taken by the body in reaching the bottom of the plane.
Take mu k=0.2 and g=10m/s2
Fp = Mg*sin30 = M*10*sin30 = 5M. = Force parallel to the incline in Newtons.
Fn = M*g*cos30 = M*10*Cos30 = 8.66M = Normal force = Force perpendicular to the incline.
Fk = u*Fn = 0.2 * 8.66M = 1.732M = Force of kinetic friction.
Fp-Fk = M*a.
5M-1.732M = M*a
3.268M = M*a
a = 3.268 m/8s^2
d = Vo*t + 0.5a*t^2 = 6.4.
0*t + 0.5*3.268t^2 = 6.4
1.634t^2 = 6.4
t = 1.98 s.
To find the time taken by the body to reach the bottom of the plane, we can use the principles of motion and physics. Here are the steps to find the time:
1. Split the gravitational force into two components:
- The component parallel to the inclined plane: F_parallel = m * g * sin(30°)
- The component perpendicular to the inclined plane: F_perpendicular = m * g * cos(30°)
2. Calculate the frictional force acting on the body:
- The frictional force is given by F_friction = μ * F_perpendicular, where μ is the coefficient of friction (given as 0.2)
3. Calculate the net force acting on the body:
- The net force is the difference between the gravitational force and the frictional force:
F_net = F_parallel - F_friction
4. Apply Newton's second law of motion:
- The net force is equal to the mass times acceleration: F_net = m * a
- Rearrange the equation to solve for acceleration: a = F_net / m
5. Use the kinematic equation to find the time taken:
- Since the body starts from rest, we can use the equation v = u + a * t, where u = 0 (initial velocity) and v = 6.4m (final velocity at the bottom of the plane).
- Rearrange the equation to solve for time: t = (v - u) / a
Now, let's plug in the given values and calculate:
1. F_parallel = m * g * sin(30°)
= m * 10 m/s^2 * 0.5
= 5m * 10 m/s^2
= 50 m/s^2
2. F_perpendicular = m * g * cos(30°)
= m * 10 m/s^2 * √(3)/2
= 5m * 5√3 m/s^2
= 25√3 m/s^2
3. F_friction = μ * F_perpendicular
= 0.2 * 25√3 m/s^2
= 5√3 m/s^2
4. F_net = F_parallel - F_friction
= 50 m/s^2 - 5√3 m/s^2
= 50 - 5√3 m/s^2
≈ 40.443 m/s^2 (rounded to three decimal places)
5. a = F_net / m
= 40.443 m/s^2 / m
≈ 40.443 m/s^2 (rounded to three decimal places)
6. t = (v - u) / a
= 6.4 m / (40.443 m/s^2)
≈ 0.158 s (rounded to three decimal places)
Therefore, the body takes approximately 0.158 seconds to reach the bottom of the plane.