To solve this problem, we need to break it down into two parts: finding the acceleration of the body up the inclined plane and then using that acceleration to find the time taken to reach the top and the final speed.
1. Finding acceleration up the inclined plane:
We can use the forces acting on the body to find the acceleration up the plane. The forces acting on the body are:
- Weight (mg) acting vertically downwards
- Normal force (N) acting perpendicular to the plane
- Friction force (f) acting parallel to the plane in the opposite direction to the motion
We need to resolve the weight and applied force into components parallel and perpendicular to the plane.
- Component of weight (mg sin 30) parallel to the plane = 5 x 9.8 x sin 30 = 24.5 N
- Component of applied force (35√3N) parallel to the plane = 35√3 x cos 30 = 30N
- Component of weight (mg cos 30) perpendicular to the plane = 5 x 9.8 x cos 30 = 42.4 N
- Normal force (N) = 42.4 N
- Friction force (f) = coefficient of friction (1/2√) x Normal force = (1/2√) x 42.4 = 7.4 N
The net force parallel to the plane = Applied force - Friction force - Component of weight
= 30 - 7.4 - 24.5 = -1.9 N (negative because it is acting in the opposite direction)
Using F=ma, we can find the acceleration up the plane:
- Net force parallel to the plane = mass x acceleration
- Acceleration = Net force parallel to the plane / mass = -1.9 / 5 = -0.38 m/s^2
2. Finding time taken to reach the top and final speed:
We can use the SUVAT equations (equations of motion with constant acceleration) to find the time taken to reach the top and final speed. We assume that the body starts from rest.
Using the SUVAT equation v^2 = u^2 + 2as, we can find the final speed (v):
- u (initial velocity) = 0
- a (acceleration) = -0.38 m/s^2 (negative because it is slowing down)
- s (displacement) = 6.3m (length of the inclined plane)
- v^2 = 0^2 + 2 x (-0.38) x 6.3
- v^2 = -4.788
- v = sqrt(-4.788) (taking the square root of a negative number gives an imaginary number, which means the body does not reach the top)
Since the body does not reach the top, we cannot use the SUVAT equations to find the time taken to reach the top. Instead, we can use the equation v = u + at to find the time taken for the body to come to a stop (v=0):
- u = 0
- a = -0.38 m/s^2
- t = (v-u)/a = (0-0)/-0.38 = 0 seconds
Therefore, the time taken for the body to come to a stop is 0 seconds.
Final answer: The body does not reach the top of the inclined plane and comes to a stop before reaching it. The time taken for the body to come to a stop is 0 seconds and its final velocity is 0 m/s.
A body of mass 5kg is initially at rest at the bottom of a rough inclined plane of length 6.3m. The plane is inclined at 300 to the horizontal and coefficient of friction between the body and the plane is 1 2√ . A constant horizontal force of 35 √ 3N is applied to the body causing it to 3 accelerate up the plane. Find the time taken for the body to reach the top and its speed on
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