A loaded penguin sled weighing rests on a plane inclined at angle to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is , and the coefficient of kinetic friction is . (a) What is the least magnitude of the force parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude that will start the sled moving up the plane? (c) What value of is required to move the sled up the plane at constant velocity?
(a) The least magnitude of force that prevents the sled moving down is
(b) The minimum magnitude of F that will start the sled moving up the plane is
(c) The value of F, that is required to move the sled up the plane at constant velocity is
Coefficient of static friction,
Coefficient of kinetic friction,
The problem is based on Newton’s second law of motion which states that the rate of change of momentum of a body is equal in both magnitude and direction of the force acting on it. Use the Newton's 2nd law of motion along vertical and horizontal direction.
where, F is the net force, m is mass and a is an acceleration.
Free body Diagram of sled over the inclined plane:
First, from the weight, calculate the mass of the system,
By using Newton’s 2nd law along vertical direction (along y),
since, block is not moving upward,
Relation between static frictional force and normal force is,
When sled is moving down frictional force is upward,
Now, applying Newton’s 2nd along the x direction, for fig. a,
since, block is not moving, localid="1654149316961"
By using equation (i) in above equation,
Thus, the least magnitude of force that prevents the sled moving down is
To find the F to start the sled moving up is, again we have to use Newton’s 2nd law along y and x direction, but in this case as the sled moving up frictional force is downward.So, refer to fig. b.
Along vertical direction (y),
since block is not moving upward,
Then static frictional force,
Along horizontal direction (x),
since, block is not moving,
Hence, the minimum magnitude of F that will start the sled moving up the plane is .
To find to move the sled up the plane at constant velocity, use kinetic frictional force,
Along horizontal direction (x),
since, block is moving with constant velocity,
Hence, the value of F, that is required to move the sled up the plane at constant velocity is
A four-person bobsled comes down a straightaway at the start of a bobsled run. The straightaway is 80.0 m long and is inclined at a constant angle of with the horizontal. Assume that the combined effects of friction and air drag produce on the bobsled a constant force of 62.0 N that acts parallel to the incline and up the incline. Answer the following questions to three significant digits.
(a) If the speed of the bobsled at the start of the run is 6.20 m/s, how long does the bobsled take to come down the straightaway?
(b) Suppose the crew is able to reduce the effects of friction and air drag to 42.0 N. For the same initial velocity, how long does the bobsled now take to come down the straightaway?
A ski that is placed on snow will stick to the snow. However, when the ski is moved along the snow, the rubbing warms and partially melts the snow, reducing the coefficient of kinetic friction and promoting sliding. Waxing the ski makes it water repellent and reduces friction with the resulting layer of water. A magazine reports that a new type of plastic ski is especially water repellent and that, on a gentle 200 m slope in the Alps, a skier reduced his top-to-bottom time from 61 s with standard skis to 42 s with the new skis. Determine the magnitude of his average acceleration with (a) the standard skis and (b) the new skis. Assuming a slope, compute the coefficient of kinetic friction for (c) the standard skis and (d) the new skis.
In Fig. 6-23, a sled is held on an inclined plane by a cord pulling directly up the plane. The sled is to be on the verge of moving up the plane. In Fig. 6-28, the magnitude required of the cord’s force on the sled is plotted versus a range of values for the coefficient of static friction between sled and plane: , , and . At what angle is the plane inclined?
An amusement park ride consists of a car moving in a vertical circle on the end of a rigid boom of negligible mass. The combined weight of the car and riders is , and the circle’s radius is . At the top of the circle, what are the
(a) magnitude and
(b) direction (up or down) of the force on the car from the boom if the car’s speed is ?
What are (c) and
(d) the direction if ?
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